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Chain rule practice problems

Derivatives of compositions: differentiate the outside, keep the inside, multiply by the inside's derivative. 422 problems with worked solutions; in 421 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. How we verify

Problem 2.187

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \).
  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x}\right)}{2} \]
    productApply the product rule to the remaining terms.✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(e^{x} \cos{\left(x + \frac{\pi}{4} \right)} + e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate exp(x).✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- e^{x} \sin{\left(x + \frac{\pi}{4} \right)} + e^{x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    trig algebraDifferentiate cos(x + pi/4). Simplify the expression inside the parentheses.✓ Proved
  6. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2} \]
    algebraFactor out exp(x).✓ Proved
  7. \[ = e^{x} \sin{\left(x \right)} \]
    rewrite algebra algebra simplify algebra algebra simplifyExpand the trigonometric terms using angle addition formulas. Substitute the value of cos(pi/4) and sin(pi/4). Distribute the values. Combine like terms inside the parentheses. Simplify the coefficient of sin(x). Multiply the negative signs. Simplify the final constant.✓ Proved
Answer \( e^{x} \sin{\left(x \right)} \)
Top: the graph of f = -sqrt(2)·e^(x)·cos(x + pi/4)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(2)·e^(x)·sin(x + pi/4)/2 - sqrt(2)·e^(x)·cos(x + pi/4)/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}\right) \]
    derivative rewritePull out the constant factor -sqrt(2)/2✓ Proved
  2. \[ = - \frac{\sqrt{2} \left(e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x}\right)}{2} \]
    productApply the product rule to exp(x)*cos(x + pi/4)✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{x} \cos{\left(x + \frac{\pi}{4} \right)} + e^{x} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate exp(x) to get exp(x)✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(- e^{x} \sin{\left(x + \frac{\pi}{4} \right)} + e^{x} \cos{\left(x + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate cos(x + pi/4) to get -sin(x + pi/4)✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2} \]
    algebra simplifyFactor out exp(x) and combine terms Final simplified derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.6

hard Beautiful Video
Differentiate \( \displaystyle f(x) = x \ln{\left(4 x + 2 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2} \).
  1. \[ \frac{d}{d x} \left(x \ln{\left(4 x + 2 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} \]
    sumApply the sum rule to separate the terms.✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    derivativeDifferentiate the term x.✓ Proved
  4. \[ = x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    chain algebraApply the chain rule to log(4*x + 2). Multiply the numerator.✓ Proved
  6. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} + \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 1 \]
    constant-multiplePull out the constant 1/2.✓ Proved
  7. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} - 1 + \frac{1}{2 x + 1} \]
    chain algebraApply the chain rule to log(2*x + 1). Simplify the constant multiplication.✓ Proved
  8. \[ = \frac{2 x}{2 x + 1} + \ln{\left(4 x + 2 \right)} - 1 + \frac{1}{2 x + 1} \]
    algebraSimplify the fraction (4*x)/(4*x + 2) by dividing by 2.✓ Proved
  9. \[ = \ln{\left(4 x + 2 \right)} \]
    algebra simplify simplifyCombine the fractions with the same denominator. Simplify the fraction (2*x + 1)/(2*x + 1). Combine the constant terms.✓ Proved
Answer \( \log{\left(4 x + 2 \right)} \)
Top: the graph of f = x·log(4x + 2) - x + log(2x + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x/(4x + 2) + log(4x + 2) - 1 + 1/(2x + 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(x \ln{\left(4 x + 2 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} \]
    sum✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    constant✓ Proved
  4. \[ = x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    product✓ Proved
  5. \[ = x \frac{d}{d x} \ln{\left(4 x + 2 \right)} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    constant-multiple simplify✓ Proved
  6. \[ = \frac{4 x}{4 x + 2} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    chain simplifydifferentiate log(4*x+2) simplify fraction✓ Proved
  7. \[ = \frac{2 x}{2 x + 1} + \ln{\left(4 x + 2 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x + 1 \right)}}{2} - 1 \]
    simplify simplifyreduce factor 4/2 combine x terms✓ Proved
  8. \[ = \frac{2 x}{2 x + 1} + \ln{\left(4 x + 2 \right)} + \frac{\frac{d}{d x} \ln{\left(2 x + 1 \right)}}{2} - 1 \]
    constant-multiplepull out 1/2✓ Proved
  9. \[ = \frac{2 x}{2 x + 1} + \ln{\left(4 x + 2 \right)} - 1 + \frac{1}{2 x + 1} \]
    chain simplify simplify sumdifferentiate log(2*x+1) simplify multiply 1/2*2 combine terms✓ Proved
  10. \[ = \ln{\left(4 x + 2 \right)} \]
    algebra simplify simplifyadd fractions cancel 1 final result✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.205

hard Beautiful Video
Differentiate \( \displaystyle f(x) = x \ln{\left(4 x - 3 \right)} - x - \frac{3 \ln{\left(4 x - 3 \right)}}{4} \).
  1. \[ \frac{d}{d x} \left(x \ln{\left(4 x - 3 \right)} - x - \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} \]
    sumApply the sum rule for derivatives.✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(4 x - 3 \right)}}{4} - 1 \]
    constantThe derivative of x is 1.✓ Proved
  4. \[ = \frac{d}{d x} x \ln{\left(4 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
    constant-multiplePull out the constant factor 3/4.✓ Proved
  5. \[ = x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} x - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
    productApply the product rule to the first term.✓ Proved
  6. \[ = x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
    derivative algebraThe derivative of x is 1. Rearrange the terms.✓ Proved
  7. \[ = \left(x - \frac{3}{4}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} - 1 \]
    algebraGroup the terms containing the derivative.✓ Proved
  8. \[ = \frac{\left(x - \frac{3}{4}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} + \ln{\left(4 x - 3 \right)} - 1 \]
    chainApply the chain rule to the logarithm.✓ Proved
  9. \[ = \ln{\left(4 x - 3 \right)} \]
    derivative algebra algebra simplifyThe derivative of 4*x - 3 is 4. Distribute the 4 in the numerator. Simplify the fraction. Final simplification.✓ Proved
Answer \( \log{\left(4 x - 3 \right)} \)
Top: the graph of f = x·log(4x - 3) - x - 3log(4x - 3)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x/(4x - 3) + log(4x - 3) - 1 - 3/(4x - 3).
A second solution, written separately

By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(x \ln{\left(4 x - 3 \right)} - x - \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) \]
    derivativeStart by expressing the derivative of the given function with respect to x.Not checked
  2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} x \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \left(- \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) \]
    sumApply the sum rule to differentiate each term separately.✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \left(- \frac{3 \ln{\left(4 x - 3 \right)}}{4}\right) - 1 \]
    constant-multipleThe derivative of -x is -1, and the constant factor -3/4 can be pulled out of the last term's derivative.✓ Proved
  4. \[ = x \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{4} - 1 \]
    productApply the product rule to the first term x*log(4*x - 3).✓ Proved
  5. \[ = \frac{x \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} + \ln{\left(4 x - 3 \right)} - 1 - \frac{3 \frac{d}{d x} \left(4 x - 3\right)}{4 \left(4 x - 3\right)} \]
    chainApply the chain rule to differentiate log(4*x - 3).✓ Proved
  6. \[ = \frac{4 x}{4 x - 3} + \ln{\left(4 x - 3 \right)} - 1 - \frac{3}{4 x - 3} \]
    chain simplify algebraDifferentiate the inner function 4*x - 3 to get 4. Simplify the fractions: x*4/(4x-3) becomes 4x/(4x-3) and 3/4 * 4/(4x-3) becomes 3/(4x-3). Remove parentheses for clarity.✓ Proved
  7. \[ = \ln{\left(4 x - 3 \right)} \]
    algebra simplify simplifyCombine the fractional terms: 4x/(4x-3) - 3/(4x-3) = (4x-3)/(4x-3). Simplify the fraction (4x-3)/(4x-3) to 1. Combine the constants: -1 + 1 = 0.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.230

hard Beautiful Video
Differentiate \( \displaystyle f(x) = x \ln{\left(5 x + 2 \right)} - x + \frac{2 \ln{\left(5 x + 2 \right)}}{5} \).
  1. \[ \frac{d}{d x} \left(x \ln{\left(5 x + 2 \right)} - x + \frac{2 \ln{\left(5 x + 2 \right)}}{5}\right) \]
    sumStart with the derivative of the entire function.✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 2 \right)}}{5} \]
    sumDistribute the derivative over the sum.✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 2 \right)}}{5} - 1 \]
    constantThe derivative of x is 1.✓ Proved
  4. \[ = \frac{d}{d x} x \ln{\left(5 x + 2 \right)} + \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 1 \]
    constant-multiplePull out the constant factor 2/5.✓ Proved
  5. \[ = x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} x + \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 1 \]
    productApply the product rule to the first term.✓ Proved
  6. \[ = x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} + \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 1 \]
    constant algebraThe derivative of x is 1. Simplify the multiplication by 1.✓ Proved
  7. \[ = \left(x + \frac{2}{5}\right) \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} - 1 \]
    algebraGroup the terms involving the derivative of the logarithm.✓ Proved
  8. \[ = \frac{\left(x + \frac{2}{5}\right) \frac{d}{d x} \left(5 x + 2\right)}{5 x + 2} + \ln{\left(5 x + 2 \right)} - 1 \]
    logarithmicApply the chain rule for the derivative of the logarithm.✓ Proved
  9. \[ = \ln{\left(5 x + 2 \right)} \]
    derivative algebra simplify simplifyThe derivative of 5*x + 2 is 5. Distribute the 5 into the parentheses. Simplify the fraction. Final simplification.✓ Proved
Answer \( \log{\left(5 x + 2 \right)} \)
Top: the graph of f = x·log(5x + 2) - x + 2log(5x + 2)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(5x + 2) + log(5x + 2) - 1 + 2/(5x + 2).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(x \ln{\left(5 x + 2 \right)} - x + \frac{2 \ln{\left(5 x + 2 \right)}}{5}\right) \]
    derivative rewriteCombine constants✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x + 2 \right)} + \frac{d}{d x} \frac{2 \ln{\left(5 x + 2 \right)}}{5} \]
    sumSeparate terms✓ Proved
  3. \[ = x \frac{d}{d x} \ln{\left(5 x + 2 \right)} + \ln{\left(5 x + 2 \right)} \frac{d}{d x} x + \frac{2 \frac{d}{d x} \ln{\left(5 x + 2 \right)}}{5} - 1 \]
    productApply product rule to x*log✓ Proved
  4. \[ = \frac{5 x}{5 x + 2} + \ln{\left(5 x + 2 \right)} - 1 + \frac{2}{5 x + 2} \]
    chain simplifyDifferentiate log(5*x+2) Simplify fractions✓ Proved
  5. \[ = \ln{\left(5 x + 2 \right)} \]
    algebra simplify simplifyCombine fractions Cancel terms Final derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.261

hard Beautiful Video
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)} \).
  1. \[ \frac{d}{d x} \left(5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} 5 x + \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} \]
    sumApply the sum rule for derivatives.✓ Proved
  3. \[ = \frac{d}{d x} 5 x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
    constantThe derivative of 5*x is 5.✓ Proved
  4. \[ = 5 \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
    constant-multipleFactor out the constant 5 from the first term.✓ Proved
  5. \[ = 5 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 5 \ln{\left(x - 3 \right)} \frac{d}{d x} x - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
    productApply the product rule to the first term.✓ Proved
  6. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - \frac{d}{d x} 15 \ln{\left(x - 3 \right)} - 5 \]
    chain algebraThe derivative of log(x - 3) is 1/(x - 3). Simplify the expression inside the parentheses.✓ Proved
  7. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - 5 - \frac{15}{x - 3} \]
    chainThe derivative of 15*log(x - 3) is 15/(x - 3).✓ Proved
  8. \[ = 5 \ln{\left(x - 3 \right)} \]
    algebraCombine all terms over a common denominator.✓ Proved
  9. \[ = \frac{5 x \ln{\left(x - 3 \right)} - 15 \ln{\left(x - 3 \right)}}{x - 3} \]
    algebra simplifyExpand the numerator. Simplify the numerator by canceling terms.✓ Proved
  10. \[ = 5 \ln{\left(x - 3 \right)} \]
    algebra simplifyFactor out the common term 15*log(x - 3) is not correct, let's factor out 5*log(x-3). Cancel the common factor (x - 3).✓ Proved
Answer \( 5 \log{\left(x - 3 \right)} \)
Top: the graph of f = 5x·log(x - 3) - 5x - 15log(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(x - 3) + 5log(x - 3) - 5 - 15/(x - 3).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(5 x \ln{\left(x - 3 \right)} - 5 x - 15 \ln{\left(x - 3 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - 5 \frac{d}{d x} x + 5 \frac{d}{d x} x \ln{\left(x - 3 \right)} - 15 \frac{d}{d x} \ln{\left(x - 3 \right)} \]
    productApply linearity of derivative✓ Proved
  3. \[ = 5 x \frac{d}{d x} \ln{\left(x - 3 \right)} + 5 \ln{\left(x - 3 \right)} \frac{d}{d x} x - 15 \frac{d}{d x} \ln{\left(x - 3 \right)} - 5 \]
    productDifferentiate the product x*log(x-3)✓ Proved
  4. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - 15 \frac{d}{d x} \ln{\left(x - 3 \right)} - 5 \]
    chain simplifyDerivative of log(x-3) is 1/(x-3) Distribute the 5✓ Proved
  5. \[ = \frac{5 x}{x - 3} + 5 \ln{\left(x - 3 \right)} - 5 - \frac{15}{x - 3} \]
    chain simplifyDifferentiate log(x-3) again Combine terms✓ Proved
  6. \[ = 5 \ln{\left(x - 3 \right)} - 5 + \frac{5 x - 15}{x - 3} \]
    simplify simplifyCombine fractions Factor 5 from numerator✓ Proved
  7. \[ = 5 \ln{\left(x - 3 \right)} \]
    simplify simplifyCancel (x-3)/(x-3) Final simplified derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.358

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \).
  1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
    constant-multipleFactor out the constant.✓ Proved
  4. \[ = - \frac{\sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{d}{d x} \frac{x}{2} \]
    productApply the product rule.✓ Proved
  5. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
    chain algebraApply the chain rule to the trigonometric terms. Simplify the expression inside the parentheses.✓ Proved
  6. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{1}{2} \]
    derivative constant-multiple algebraEvaluate the derivative of x/2. Distribute the 1/8. Factor out 1/2.✓ Proved
  7. \[ = \frac{1}{2} - \frac{\cos{\left(8 x \right)}}{2} \]
    trig algebra algebraUse the double angle identity cos(2*theta) = cos^2(theta) - sin^2(theta). Simplify the argument of the cosine function. Factor out 1/2.✓ Proved
  8. \[ = \sin^{2}{\left(4 x \right)} \]
    simplifyUse the identity 1 - cos(2*theta) = 2*sin(theta)**2 with theta = 4*x.✓ Proved
Answer \( \sin^{2}{\left(4 x \right)} \)
Top: the graph of f = x/2 - sin(4x)·cos(4x)/8. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sin(4x)²/2 - cos(4x)²/2 + 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8}\right) \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
    sumSeparate the derivative of each term.✓ Proved
  3. \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \]
    constant-multiplePull out the constant 1/8.✓ Proved
  4. \[ = - \frac{\sin{\left(4 x \right)} \frac{d}{d x} \cos{\left(4 x \right)}}{8} - \frac{\cos{\left(4 x \right)} \frac{d}{d x} \sin{\left(4 x \right)}}{8} + \frac{d}{d x} \frac{x}{2} \]
    productApply product rule to sin(4*x)*cos(4*x).✓ Proved
  5. \[ = \frac{\sin^{2}{\left(4 x \right)} \frac{d}{d x} 4 x}{8} - \frac{\cos^{2}{\left(4 x \right)} \frac{d}{d x} 4 x}{8} + \frac{d}{d x} \frac{x}{2} \]
    chainDifferentiate sin(4*x) and cos(4*x) using chain rule.✓ Proved
  6. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]
    derivative simplify simplify simplifyDerivative of 4*x is 4. Combine terms. Factor out 4. Divide numerator and denominator by 4.✓ Proved
  7. \[ = \frac{\sin^{2}{\left(4 x \right)}}{2} - \frac{\cos^{2}{\left(4 x \right)}}{2} + \frac{1}{2} \]
    derivative simplify simplify simplifyDerivative of x/2 is 1/2. Combine over common denominator. Distribute the minus sign. Reorder terms.✓ Proved
  8. \[ = \sin^{2}{\left(4 x \right)} \]
    trig simplifyUse identity 1 - cos^2 = sin^2. Combine like terms.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.1034

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \).
  1. \[ \frac{d}{d x} \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    chainApply the chain rule for the arctangent function.✓ Proved
  3. \[ = \frac{\frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    sumApply the sum rule to the inner derivative.✓ Proved
  4. \[ = \frac{- 3 \csc^{2}{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    derivativeDifferentiate the cotangent term.✓ Proved
  5. \[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} - 3 \csc^{2}{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    derivativeDifferentiate the cosecant term.✓ Proved
  6. \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    algebra algebraFactor out common terms in the numerator. Combine the expression into a single fraction.✓ Proved
  7. \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{\cot^{2}{\left(3 x \right)} + 2 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + \csc^{2}{\left(3 x \right)} + 1} \]
    algebraExpand the squared term in the denominator.✓ Proved
  8. \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{2 \cot{\left(3 x \right)} \csc{\left(3 x \right)} + 2 \csc^{2}{\left(3 x \right)}} \]
    algebra simplifyUse the identity cot(3*x)**2 = csc(3*x)**2 - 1. Simplify the denominator by combining like terms.✓ Proved
  9. \[ = - \frac{3}{2} \]
    algebra simplify simplifyFactor the denominator. Cancel the common factor (csc(3*x) + cot(3*x)). Cancel the common factor csc(3*x) to get the final result.✓ Proved
Answer \( - \frac{3}{2} \)
Top: the graph of f = atan(cot(3x) + csc(3x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (-3cot(3x)² - 3cot(3x)·csc(3x) - 3)/((cot(3x) + csc(3x))² + 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \operatorname{atan}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)} \right)} \]
    derivative✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    chain✓ Proved
  3. \[ = \frac{\frac{d}{d x} \cot{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    sum✓ Proved
  4. \[ = \frac{- 3 \csc^{2}{\left(3 x \right)} + \frac{d}{d x} \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    derivative✓ Proved
  5. \[ = \frac{- 3 \cot{\left(3 x \right)} \csc{\left(3 x \right)} - 3 \csc^{2}{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    derivative algebra✓ Proved
  6. \[ = - \frac{3 \left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right) \csc{\left(3 x \right)}}{\left(\cot{\left(3 x \right)} + \csc{\left(3 x \right)}\right)^{2} + 1} \]
    simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.135

hard Beautiful Video
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \).
  1. \[ \frac{d}{d x} \left(2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)}\right) \]
    sumStart with the derivative of the entire expression.✓ Proved
  2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} \]
    sumApply the sum rule to separate the terms.✓ Proved
  3. \[ = \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    constantThe derivative of 2*x is 2.✓ Proved
  4. \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + 2 \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    derivativeThe derivative of 2*x is 2.✓ Proved
  6. \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} + 4 \frac{d}{d x} \ln{\left(x + 2 \right)} - 2 \]
    logarithmicThe derivative of log(x + 2) is 1/(x + 2).✓ Proved
  7. \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} - 2 + \frac{4}{x + 2} \]
    derivative algebraThe derivative of 4*log(x + 2) is 4/(x + 2). Distribute the constant in the last term.✓ Proved
  8. \[ = 2 \ln{\left(x + 2 \right)} - 2 + \frac{2 x + 4}{x + 2} \]
    algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
  9. \[ = 2 \ln{\left(x + 2 \right)} \]
    simplify simplifyCancel the common factor (x + 2). Simplify the remaining constant terms.✓ Proved
Answer \( 2 \log{\left(x + 2 \right)} \)
Top: the graph of f = 2x·log(x + 2) - 2x + 4log(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x/(x + 2) + 2log(x + 2) - 2 + 4/(x + 2).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)}\right) \]
    derivative✓ Proved
  2. \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + 2 \ln{\left(x + 2 \right)} \frac{d}{d x} x + 4 \frac{d}{d x} \ln{\left(x + 2 \right)} - 2 \]
    product✓ Proved
  3. \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} - 2 + \frac{4}{x + 2} \]
    derivative algebra✓ Proved
  4. \[ = 2 \ln{\left(x + 2 \right)} - 2 + \frac{2 x + 4}{x + 2} \]
    algebra✓ Proved
  5. \[ = 2 \ln{\left(x + 2 \right)} \]
    algebra simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.348

hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{\sqrt{1 - x^{2}}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{1 - x^{2}}} \right)} \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \ln{\left(\sqrt{1 - x^{2}} \right)}\right) \]
    logarithmic algebraUse the property log(a/b) = log(a) - log(b). Rewrite the square root as a power.✓ Proved
  3. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \frac{\ln{\left(1 - x^{2} \right)}}{2}\right) \]
    constant-multiplePull out the constant factor 1/2.✓ Proved
  4. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(1 - x^{2} \right)}}{2} \]
    sumApply the difference rule for derivatives.✓ Proved
  5. \[ = - \frac{d}{d x} \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{1}{x} \]
    derivativeDifferentiate the first term.✓ Proved
  6. \[ = - \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{1}{x} \]
    constant-multiplePull out the constant factor 1/2 from the second term.✓ Proved
  7. \[ = - \frac{\frac{d}{d x} \left(1 - x^{2}\right)}{2 \left(1 - x^{2}\right)} + \frac{1}{x} \]
    chainApply the chain rule to the logarithm.✓ Proved
  8. \[ = \frac{x}{1 - x^{2}} + \frac{1}{x} \]
    derivative algebraDifferentiate the inner function 1 - x**2. Simplify the signs and constants.✓ Proved
  9. \[ = \frac{1}{x \left(1 - x^{2}\right)} \]
    algebra simplifyFind a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{1}{- x^{3} + x} \)
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{1 - x^{2}}} \right)} \]
    derivative✓ Proved
  2. \[ = \frac{\sqrt{1 - x^{2}} \frac{d}{d x} \frac{x}{\sqrt{1 - x^{2}}}}{x} \]
    chain rewriteDerivative of log(u) is u'/u Rewrite the quotient as a product with a negative exponent✓ Proved
  3. \[ = \frac{\sqrt{1 - x^{2}} \left(\frac{x^{2}}{\left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{1 - x^{2}}}\right)}{x} \]
    product simplifyDifferentiate the product x*(1 - x**2)**(-1/2) Simplify the derivative of the second factor✓ Proved
  4. \[ = \frac{1}{x \left(1 - x^{2}\right)} \]
    simplify simplify simplifyCombine terms in the numerator Subtract exponents and simplify Final simplified derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.531

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \]
    derivative algebraStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
  2. \[ = \frac{d}{d x} \left(x + \ln{\left(\frac{1}{\cos{\left(x + 1 \right)}} \right)} + 1\right) \]
    logarithmic logarithmicApply the logarithm product rule. Simplify log(exp(x + 1)).✓ Proved
  3. \[ = \frac{d}{d x} \left(x - \ln{\left(\cos{\left(x + 1 \right)} \right)} + 1\right) \]
    logarithmic constantApply the logarithm quotient rule. Simplify log(1) to 0.✓ Proved
  4. \[ = \frac{d}{d x} \left(x + 1\right) - \frac{d}{d x} \ln{\left(\cos{\left(x + 1 \right)} \right)} \]
    sumApply the difference rule for derivatives.✓ Proved
  5. \[ = 1 - \frac{d}{d x} \ln{\left(\cos{\left(x + 1 \right)} \right)} \]
    derivativeDifferentiate the first term.✓ Proved
  6. \[ = 1 - \frac{\frac{d}{d x} \cos{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \frac{\sin{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} + 1 \]
    trig algebraDifferentiate the cosine function. Simplify the signs and the fraction.✓ Proved
  8. \[ = \tan{\left(x + 1 \right)} + 1 \]
    trigUse the identity tan(u) = sin(u)/cos(u).✓ Proved
Answer \( \tan{\left(x + 1 \right)} + 1 \)
Top: the graph of f = log(e^(x + 1)/cos(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (e^(x + 1)·sin(x + 1)/cos(x + 1)² + e^(x + 1)/cos(x + 1))·e^(-x - 1)·cos(x + 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \right)} \]
    derivative✓ Proved
  2. \[ = e^{- x - 1} \cos{\left(x + 1 \right)} \frac{d}{d x} \frac{e^{x + 1}}{\cos{\left(x + 1 \right)}} \]
    chain✓ Proved
  3. \[ = \frac{\left(- e^{x + 1} \frac{d}{d x} \cos{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \frac{d}{d x} e^{x + 1}\right) e^{- x - 1}}{\cos{\left(x + 1 \right)}} \]
    quotient✓ Proved
  4. \[ = \frac{\left(e^{x + 1} \sin{\left(x + 1 \right)} + e^{x + 1} \cos{\left(x + 1 \right)}\right) e^{- x - 1}}{\cos{\left(x + 1 \right)}} \]
    derivative✓ Proved
  5. \[ = \frac{\left(\sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)}\right) e^{- x - 1} e^{x + 1}}{\cos{\left(x + 1 \right)}} \]
    algebra algebra✓ Proved
  6. \[ = \frac{\sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}} \]
    simplify✓ Proved
  7. \[ = \tan{\left(x + 1 \right)} + 1 \]
    simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.1143

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(2 x - \ln{\left(4 x^{2} \right)}\right) \]
    algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(2*x)) to 2*x.✓ Proved
  3. \[ = \frac{d}{d x} 2 x - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
    sumSplit the derivative using the difference rule.✓ Proved
  4. \[ = 2 - \frac{d}{d x} \ln{\left(4 x^{2} \right)} \]
    constantThe derivative of 2*x is 2.✓ Proved
  5. \[ = 2 - \frac{d}{d x} \left(\ln{\left(x^{2} \right)} + \ln{\left(4 \right)}\right) \]
    algebraUse the property log(ab) = log(a) + log(b).✓ Proved
  6. \[ = - \frac{d}{d x} \ln{\left(4 \right)} - \frac{d}{d x} \ln{\left(x^{2} \right)} + 2 \]
    sumDistribute the derivative.✓ Proved
  7. \[ = 2 - \frac{d}{d x} \ln{\left(x^{2} \right)} \]
    constantThe derivative of the constant log(4) is 0.✓ Proved
  8. \[ = 2 - \frac{d}{d x} 2 \ln{\left(x \right)} \]
    rewriteUse the power rule for logarithms: log(x**2) = 2*log(x).✓ Proved
  9. \[ = 2 - 2 \frac{d}{d x} \ln{\left(x \right)} \]
    constant-multiplePull the constant factor 2 out of the derivative.✓ Proved
  10. \[ = 2 - \frac{2}{x} \]
    derivative simplifyThe derivative of log(x) is 1/x. Simplify the expression.✓ Proved
Answer \( 2 - \frac{2}{x} \)
Top: the graph of f = log(e^(2x)/(4x²)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x²·(e^(2x)/(2x²) - e^(2x)/(2x³))·e^(-2x).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{2 x}}{4 x^{2}} \right)} \]
    derivative✓ Proved
  2. \[ = 4 x^{2} e^{- 2 x} \frac{d}{d x} \frac{e^{2 x}}{4 x^{2}} \]
    chain✓ Proved
  3. \[ = \frac{\left(8 x^{2} e^{2 x} - 8 x e^{2 x}\right) e^{- 2 x}}{4 x^{2}} \]
    quotient✓ Proved
  4. \[ = \frac{8 x^{2} - 8 x}{4 x^{2}} \]
    algebra✓ Proved
  5. \[ = \frac{2 \left(x - 1\right)}{x} \]
    algebra algebra algebra✓ Proved
  6. \[ = \frac{2 x - 2}{x} \]
    algebra✓ Proved
  7. \[ = 2 - \frac{2}{x} \]
    simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.1260

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{\sqrt{x^{2} + 1}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{x^{2} + 1}} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \ln{\left(\sqrt{x^{2} + 1} \right)}\right) \]
    algebra algebraUse logarithm properties to split the fraction. Rewrite the square root as a fractional power.✓ Proved
  3. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]
    sumApply the difference rule for derivatives.✓ Proved
  4. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
    constant-multiplePull out the constant factor 1/2 using the power rule for logs.✓ Proved
  5. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
    constant-multipleApply the constant multiple rule.✓ Proved
  6. \[ = - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} + \frac{1}{x} \]
    derivativeDifferentiate the first term.✓ Proved
  7. \[ = - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} + \frac{1}{x} \]
    chainApply the chain rule to the second term.✓ Proved
  8. \[ = - \frac{x}{x^{2} + 1} + \frac{1}{x} \]
    derivative algebraDifferentiate the inner function x**2 + 1. Simplify the expression by canceling the 2 and the 1/2.✓ Proved
  9. \[ = \frac{1}{x \left(x^{2} + 1\right)} \]
    algebra simplifyFind a common denominator to combine the terms. Simplify the numerator.✓ Proved
Answer \( \frac{1}{x^{3} + x} \)
Top: the graph of f = log(x/sqrt(x² + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(x² + 1)·(-x²/(x² + 1)^(3/2) + 1/sqrt(x² + 1))/x.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{x^{2} + 1}} \right)} \]
    derivative✓ Proved
  2. \[ = \frac{\sqrt{x^{2} + 1} \frac{d}{d x} \frac{x}{\sqrt{x^{2} + 1}}}{x} \]
    chainApply chain rule to log✓ Proved
  3. \[ = \frac{\sqrt{x^{2} + 1} \left(- \frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} + 1}}\right)}{x} \]
    productDifferentiate x*(x**2+1)**(-1/2) using product rule✓ Proved
  4. \[ = \frac{1}{x \left(x^{2} + 1\right)} \]
    simplify simplify simplifyFactor common power Combine powers and cancel Final simplified derivative✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.90

hard Beautiful Video
Differentiate \( \displaystyle f(x) = x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)} \).
  1. \[ \frac{d}{d x} \left(x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} \]
    sumApply the sum rule for differentiation.✓ Proved
  3. \[ = \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} - 1 \]
    constantThe derivative of x is 1.✓ Proved
  4. \[ = x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} x - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} - 1 \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} - 3 \frac{d}{d x} \ln{\left(x - 3 \right)} - 1 \]
    constantThe derivative of x is 1.✓ Proved
  6. \[ = \frac{x}{x - 3} + \ln{\left(x - 3 \right)} - 1 - \frac{3}{x - 3} \]
    logarithmic constant-multipleThe derivative of log(x - 3) is 1/(x - 3). Simplify the coefficients.✓ Proved
  7. \[ = \ln{\left(x - 3 \right)} \]
    algebra simplify simplifyCombine the fractions with the same denominator. Simplify the fraction (x-3)/(x-3). Final simplification.✓ Proved
Answer \( \log{\left(x - 3 \right)} \)
Top: the graph of f = x·log(x - 3) - x - 3log(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x/(x - 3) + log(x - 3) - 1 - 3/(x - 3).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(x \ln{\left(x - 3 \right)} - x - 3 \ln{\left(x - 3 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(x - 3 \right)} - \frac{d}{d x} 3 \ln{\left(x - 3 \right)} \]
    sum✓ Proved
  3. \[ = x \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} x - 3 \frac{d}{d x} \ln{\left(x - 3 \right)} - 1 \]
    product✓ Proved
  4. \[ = \frac{x}{x - 3} + \ln{\left(x - 3 \right)} - 1 - \frac{3}{x - 3} \]
    derivative simplify✓ Proved
  5. \[ = \ln{\left(x - 3 \right)} \]
    simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.269

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \).
  1. \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)} \]
    constantPull out the negative sign.✓ Proved
  3. \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    chainApply the chain rule for the logarithm.✓ Proved
  4. \[ = - \frac{\frac{d}{d x} \cot{\left(x - 1 \right)} + \frac{d}{d x} \csc{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    sumApply the sum rule to the inner expression.✓ Proved
  5. \[ = - \frac{- \csc^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) + \frac{d}{d x} \csc{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    trigDifferentiate the cotangent term.✓ Proved
  6. \[ = - \frac{- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} - \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    derivative algebraDifferentiate the cosecant term and the inner linear term. Simplify the expression inside the parentheses.✓ Proved
  7. \[ = \frac{\cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} + \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    algebraDistribute the negative sign and the fraction.✓ Proved
  8. \[ = \csc{\left(x - 1 \right)} \]
    algebra simplifyFactor out csc(x - 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(x - 1 \right)}} \)
Top: the graph of f = -log(cot(x - 1) + csc(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-cot(x - 1)² - cot(x - 1)·csc(x - 1) - 1)/(cot(x - 1) + csc(x - 1)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)} \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    derivative✓ Proved
  3. \[ = - \frac{\frac{d}{d x} \cot{\left(x - 1 \right)} + \frac{d}{d x} \csc{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    sum✓ Proved
  4. \[ = - \frac{- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) - \csc^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    chain✓ Proved
  5. \[ = - \frac{- \cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} - \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    constant simplify✓ Proved
  6. \[ = \frac{\cot{\left(x - 1 \right)} \csc{\left(x - 1 \right)} + \csc^{2}{\left(x - 1 \right)}}{\cot{\left(x - 1 \right)} + \csc{\left(x - 1 \right)}} \]
    algebra✓ Proved
  7. \[ = \csc{\left(x - 1 \right)} \]
    algebra simplifyfactor numerator cancel common factor✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.289

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - 3 \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)} \).
  1. \[ \frac{d}{d x} \left(- 3 \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - 3 \frac{d}{d x} \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = - \frac{3 \frac{d}{d x} \left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    chainApply the chain rule for the logarithm.✓ Proved
  4. \[ = - \frac{3 \left(\frac{d}{d x} \cot{\left(x - 3 \right)} + \frac{d}{d x} \csc{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    sumDifferentiate the sum inside the parenthesis.✓ Proved
  5. \[ = - \frac{3 \left(- \cot{\left(x - 3 \right)} \csc{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right) - \csc^{2}{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    trigApply the derivatives for cotangent and cosecant.✓ Proved
  6. \[ = - \frac{3 \left(- \cot{\left(x - 3 \right)} \csc{\left(x - 3 \right)} - \csc^{2}{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    derivative algebraEvaluate the derivatives of the inner linear functions. Simplify the expression by multiplying the 1s.✓ Proved
  7. \[ = \frac{3 \cot{\left(x - 3 \right)} \csc{\left(x - 3 \right)} + 3 \csc^{2}{\left(x - 3 \right)}}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    algebraDistribute the negative sign and the constant.✓ Proved
  8. \[ = 3 \csc{\left(x - 3 \right)} \]
    algebra simplifyFactor out csc(x - 3) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{3}{\sin{\left(x - 3 \right)}} \)
Top: the graph of f = -3log(cot(x - 3) + csc(x - 3)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3(-cot(x - 3)² - cot(x - 3)·csc(x - 3) - 1)/(cot(x - 3) + csc(x - 3)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(- 3 \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)}\right) \]
    derivative✓ Proved
  2. \[ = - 3 \frac{d}{d x} \ln{\left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)} \right)} \]
    constant-multiplePull the constant -3 outside the derivative.✓ Proved
  3. \[ = - \frac{3 \frac{d}{d x} \left(\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    chainDerivative of log(u) is u' / u.✓ Proved
  4. \[ = - \frac{3 \left(\frac{d}{d x} \cot{\left(x - 3 \right)} + \frac{d}{d x} \csc{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    sumDerivative of a sum is the sum of derivatives.✓ Proved
  5. \[ = - \frac{3 \left(- \cot{\left(x - 3 \right)} \csc{\left(x - 3 \right)} - \csc^{2}{\left(x - 3 \right)}\right)}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    derivative simplifyDerivative of cot(u) is -csc(u)**2*u', derivative of csc(u) is -csc(u)*cot(u)*u'. Combine the two negative terms.✓ Proved
  6. \[ = \frac{3 \cot{\left(x - 3 \right)} \csc{\left(x - 3 \right)} + 3 \csc^{2}{\left(x - 3 \right)}}{\cot{\left(x - 3 \right)} + \csc{\left(x - 3 \right)}} \]
    simplifyFactor out the negative sign.✓ Proved
  7. \[ = 3 \csc{\left(x - 3 \right)} \]
    simplifyThe numerator and denominator cancel, leaving 3*csc(x-3).✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.906

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{1 - x}}{\cos{\left(x - 1 \right)}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{1 - x}}{\cos{\left(x - 1 \right)}} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(- x - \ln{\left(\cos{\left(x - 1 \right)} \right)} + 1\right) \]
    algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(1 - x)) to 1 - x.✓ Proved
  3. \[ = \frac{d}{d x} \left(1 - x\right) - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]
    sumApply the linearity of the derivative.✓ Proved
  4. \[ = - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} - 1 \]
    constantThe derivative of 1 - x is -1.✓ Proved
  5. \[ = -1 - \frac{\frac{d}{d x} \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]
    chainApply the chain rule to the second term.✓ Proved
  6. \[ = \frac{\sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} - 1 \]
    trig algebraThe derivative of cos(x - 1) is -sin(x - 1). Simplify the fraction.✓ Proved
  7. \[ = \tan{\left(x - 1 \right)} - 1 \]
    trig algebraUse the identity tan(u) = sin(u)/cos(u). Rearrange the terms.✓ Proved
Answer \( \tan{\left(x - 1 \right)} - 1 \)
Top: the graph of f = log(e^(1 - x)/cos(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (e^(1 - x)·sin(x - 1)/cos(x - 1)² - e^(1 - x)/cos(x - 1))·e^(x - 1)·cos(x - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{1 - x}}{\cos{\left(x - 1 \right)}} \right)} \]
    derivative✓ Proved
  2. \[ = e^{x - 1} \cos{\left(x - 1 \right)} \frac{d}{d x} \frac{e^{1 - x}}{\cos{\left(x - 1 \right)}} \]
    quotientDerivative of log(u) is u'/u✓ Proved
  3. \[ = \frac{\left(e^{1 - x} \frac{d}{d x} \sec{\left(x - 1 \right)} + \sec{\left(x - 1 \right)} \frac{d}{d x} e^{1 - x}\right) e^{x - 1}}{\sec{\left(x - 1 \right)}} \]
    productDifferentiate exp(1-x)*sec(x-1) in the numerator✓ Proved
  4. \[ = \frac{\left(- e^{1 - x} \sec{\left(x - 1 \right)} + e^{1 - x} \frac{d}{d x} \sec{\left(x - 1 \right)}\right) e^{x - 1}}{\sec{\left(x - 1 \right)}} \]
    derivativeDerivative of exp(1-x) is -exp(1-x)✓ Proved
  5. \[ = \frac{\left(e^{1 - x} \tan{\left(x - 1 \right)} \sec{\left(x - 1 \right)} - e^{1 - x} \sec{\left(x - 1 \right)}\right) e^{x - 1}}{\sec{\left(x - 1 \right)}} \]
    derivativeDerivative of sec(x-1) is sec(x-1)*tan(x-1)✓ Proved
  6. \[ = \left(\tan{\left(x - 1 \right)} - 1\right) e^{1 - x} e^{x - 1} \]
    algebraFactor out common terms✓ Proved
  7. \[ = \tan{\left(x - 1 \right)} - 1 \]
    simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.247

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \).
  1. \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)} \]
    constantPull the constant -1 out of the derivative.✓ Proved
  3. \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}\right)}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    chainApply the chain rule for the natural logarithm.✓ Proved
  4. \[ = - \frac{\frac{d}{d x} \cot{\left(x + 2 \right)} + \frac{d}{d x} \csc{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    sumApply the sum rule to the inner expression.✓ Proved
  5. \[ = - \frac{- \cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) - \csc^{2}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    trigDifferentiate the trigonometric functions using the chain rule.✓ Proved
  6. \[ = - \frac{- \cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} - \csc^{2}{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    derivativeEvaluate the derivatives of the inner linear terms.✓ Proved
  7. \[ = \csc{\left(x + 2 \right)} \]
    algebra algebra simplifyFactor out the common term -csc(x + 2). Cancel the common term in the numerator and denominator. Simplify the remaining expression.✓ Proved
Answer \( \frac{1}{\sin{\left(x + 2 \right)}} \)
Top: the graph of f = -log(cot(x + 2) + csc(x + 2)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(-cot(x + 2)² - cot(x + 2)·csc(x + 2) - 1)/(cot(x + 2) + csc(x + 2)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(- \ln{\left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)} \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{\frac{d}{d x} \left(\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}\right)}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    chainDerivative of log(u) is u' / u, with a minus sign outside.✓ Proved
  3. \[ = \frac{\cot{\left(x + 2 \right)} \csc{\left(x + 2 \right)} + \csc^{2}{\left(x + 2 \right)}}{\cot{\left(x + 2 \right)} + \csc{\left(x + 2 \right)}} \]
    derivative simplifyDerivative of cot(u) is -csc(u)**2, derivative of csc(u) is -csc(u)*cot(u). Remove the double negative.✓ Proved
  4. \[ = \csc{\left(x + 2 \right)} \]
    simplifyFactor csc(x+2) and cancel with the denominator.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.281

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4} \).
  1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(2 x \right)}}{4} \]
    sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
  3. \[ = \frac{5 \frac{d}{d x} x}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]
    constant-multipleSeparate the constants from the derivatives.✓ Proved
  4. \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]
    derivative algebraDifferentiate x. Simplify the first term.✓ Proved
  5. \[ = - \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4} + \frac{5}{2} \]
    chainApply the chain rule to sin(2*x).✓ Proved
  6. \[ = \frac{5}{2} - \frac{5 \cos{\left(2 x \right)}}{2} \]
    derivative algebraDifferentiate 2*x. Multiply the constants.✓ Proved
Answer \( 5 \sin^{2}{\left(x \right)} \)
Top: the graph of f = 5x/2 - 5sin(2x)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/2 - 5cos(2x)/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \left(\frac{5 x}{2} - \frac{5 \sin{\left(2 x \right)}}{4}\right) \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{d}{d x} \frac{5 \sin{\left(2 x \right)}}{4} \]
    sumSplit the derivative over the sum.✓ Proved
  3. \[ = \frac{d}{d x} \frac{5 x}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]
    constant-multiplePull out the constant 5/4.✓ Proved
  4. \[ = \frac{5}{2} - \frac{5 \frac{d}{d x} \sin{\left(2 x \right)}}{4} \]
    derivativeDerivative of 5*x/2 is 5/2.✓ Proved
  5. \[ = - \frac{5 \cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4} + \frac{5}{2} \]
    chainDerivative of sin(2*x) is cos(2*x) times derivative of 2*x.✓ Proved
  6. \[ = \frac{5}{2} - \frac{5 \cos{\left(2 x \right)}}{2} \]
    derivative simplifyDerivative of 2*x is 2. Multiply 5/4 by 2.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.723

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}\right)}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    chainApply the chain rule for the logarithm.✓ Proved
  3. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(x + 1 \right)}\right) + \frac{d}{d x} \csc{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    sumApply the sum rule to the inner expression.✓ Proved
  4. \[ = \frac{- \frac{d}{d x} \cot{\left(x + 1 \right)} + \frac{d}{d x} \csc{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    constantFactor out the negative sign.✓ Proved
  5. \[ = \frac{- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right) + \csc^{2}{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    trigDifferentiate the trigonometric functions using the chain rule.✓ Proved
  6. \[ = \frac{- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} + \csc^{2}{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    derivative algebraEvaluate the derivatives of the inner linear terms. Simplify the expression inside the parentheses.✓ Proved
  7. \[ = \csc{\left(x + 1 \right)} \]
    algebra simplifyFactor out csc(x + 1) from the numerator and rearrange the denominator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(x + 1 \right)}} \)
Top: the graph of f = log(-cot(x + 1) + csc(x + 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (cot(x + 1)² - cot(x + 1)·csc(x + 1) + 1)/(-cot(x + 1) + csc(x + 1)).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)} \right)} \]
    derivative✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}\right)}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    chain✓ Proved
  3. \[ = \frac{- \cot{\left(x + 1 \right)} \csc{\left(x + 1 \right)} + \csc^{2}{\left(x + 1 \right)}}{- \cot{\left(x + 1 \right)} + \csc{\left(x + 1 \right)}} \]
    derivative✓ Proved
  4. \[ = \csc{\left(x + 1 \right)} \]
    algebra simplify✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 2.908

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{x - 1}}{\cos{\left(x - 1 \right)}} \right)} \).
  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{x - 1}}{\cos{\left(x - 1 \right)}} \right)} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(x - \ln{\left(\cos{\left(x - 1 \right)} \right)} - 1\right) \]
    algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(x - 1)) to x - 1.✓ Proved
  3. \[ = \frac{d}{d x} \left(x - 1\right) - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]
    sumSplit the derivative using the difference rule.✓ Proved
  4. \[ = 1 - \frac{d}{d x} \ln{\left(\cos{\left(x - 1 \right)} \right)} \]
    derivativeThe derivative of x - 1 is 1.✓ Proved
  5. \[ = 1 - \frac{\frac{d}{d x} \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]
    chainApply the chain rule to the second term.✓ Proved
  6. \[ = \frac{\sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} + 1 \]
    trig algebraThe derivative of cos(x - 1) is -sin(x - 1). Simplify the fraction.✓ Proved
  7. \[ = \tan{\left(x - 1 \right)} + 1 \]
    trigUse the identity sin(u)/cos(u) = tan(u).✓ Proved
Answer \( \tan{\left(x - 1 \right)} + 1 \)
Top: the graph of f = log(e^(x - 1)/cos(x - 1)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (e^(x - 1)·sin(x - 1)/cos(x - 1)² + e^(x - 1)/cos(x - 1))·e^(1 - x)·cos(x - 1).
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \frac{d}{d x} \ln{\left(\frac{e^{x - 1}}{\cos{\left(x - 1 \right)}} \right)} \]
    derivative✓ Proved
  2. \[ = e^{1 - x} \cos{\left(x - 1 \right)} \frac{d}{d x} \frac{e^{x - 1}}{\cos{\left(x - 1 \right)}} \]
    logarithmic✓ Proved
  3. \[ = \frac{\left(- e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 \right)} + \cos{\left(x - 1 \right)} \frac{d}{d x} e^{x - 1}\right) e^{1 - x}}{\cos{\left(x - 1 \right)}} \]
    quotient✓ Proved
  4. \[ = \frac{\left(e^{x - 1} \sin{\left(x - 1 \right)} + e^{x - 1} \cos{\left(x - 1 \right)}\right) e^{1 - x}}{\cos{\left(x - 1 \right)}} \]
    derivative simplify✓ Proved
  5. \[ = \frac{\left(\sin{\left(x - 1 \right)} + \cos{\left(x - 1 \right)}\right) e^{1 - x} e^{x - 1}}{\cos{\left(x - 1 \right)}} \]
    algebra✓ Proved
  6. \[ = \frac{\sin{\left(x - 1 \right)} + \cos{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} \]
    simplify simplify✓ Proved
  7. \[ = \frac{\sin{\left(x - 1 \right)}}{\cos{\left(x - 1 \right)}} + 1 \]
    simplify✓ Proved
  8. \[ = \tan{\left(x - 1 \right)} + 1 \]
    trig✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More chain rule practice

Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(x \right)} \right)} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x - 3 \right)}}{2} \).
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Problem 2.62medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(5 x - 3 \right)}}{5} \).
Problem 2.64medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sin{\left(5 x + 1 \right)}}{5} \).
Problem 2.67medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 2 \right)}}{2} \).
Problem 2.70medium Video✓ Nihil obstat
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Problem 2.73medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{2 x} \).
Problem 2.74medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(4 x \right)} \right)}}{4} \).
Problem 2.76medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(3 x - 1 \right)} \).
Problem 2.79medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(5 x \right)} \right)}}{5} \).
Problem 2.82medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{5 x} \).
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Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(5 x \right)}}{5} \).
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Differentiate \( \displaystyle f(x) = - 3 \cos{\left(x - 3 \right)} \).
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Differentiate \( \displaystyle f(x) = 5 \ln{\left(\ln{\left(x + 2 \right)} \right)} \).
Problem 2.105medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \cos{\left(x - 1 \right)} \).
Problem 2.106medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x \right)}}{3} \).
Problem 2.110medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 e^{2 x + 2}}{2} \).
Problem 2.113medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \ln{\left(x + 1 \right)} \).
Problem 2.115medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x \right)}}{3} \).
Problem 2.117medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 1 \right)}}{2} \).
Problem 2.120medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.121medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{e^{4 x}}{4} \).
Problem 2.124medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.125medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x \right)}}{2} \).
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Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x \right)}}{2} \).
Problem 2.130medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\tan{\left(4 x \right)}}{4} \).
Problem 2.137medium Video✓ Nihil obstat
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Problem 2.138medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{x + 1} \).
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Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x + 1 \right)} \right)}}{2} \).
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Differentiate \( \displaystyle f(x) = 5 \ln{\left(x - 3 \right)} \).
Problem 2.160medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \cos{\left(3 x - 3 \right)}}{3} \).
Problem 2.161medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(2 x - 3 \right)} \right)}}{2} \).
Problem 2.167medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(2 x + 1 \right)} \).
Problem 2.172medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{2 x - 3} \).
Problem 2.174medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x + 2 \right)} \).
Problem 2.177medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.178medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(2 x - 1 \right)}}{2} \).
Problem 2.184medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.194medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 e^{x - 3} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x \right)} \right)}}{2} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(4 x - 1 \right)}}{4} \).
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Differentiate \( \displaystyle f(x) = 2 \sin{\left(x - 1 \right)} \).
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Differentiate \( \displaystyle f(x) = \frac{3 e^{2 x}}{2} \).
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Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(4 x \right)}}{4} \).
Problem 2.219medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 e^{x + 2} \).
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Differentiate \( \displaystyle f(x) = \frac{3 \sin{\left(2 x \right)}}{2} \).
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Differentiate \( \displaystyle f(x) = 2 \ln{\left(\ln{\left(x - 3 \right)} \right)} \).
Problem 2.225medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{e^{4 x}}{4} \).
Problem 2.229medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 e^{3 x}}{3} \).
Problem 2.232medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sin{\left(3 x - 1 \right)}}{3} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x - 1 \right)}}{5} \).
Problem 2.235medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.239medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x - 3 \right)}}{4} \).
Problem 2.241medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x - 3 \right)}}{2} \).
Problem 2.243medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.246medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(2 x + 2 \right)} \).
Problem 2.249medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \cos{\left(2 x \right)} \).
Problem 2.252medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 e^{5 x}}{5} \).
Problem 2.253medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x - 1 \right)}}{2} \).
Problem 2.257medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = e^{x + 2} \).
Problem 2.259medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sin{\left(x + 1 \right)} \).
Problem 2.262medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 e^{2 x - 1}}{2} \).
Problem 2.266medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sin{\left(5 x - 3 \right)}}{5} \).
Problem 2.267medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(2 x - 3 \right)}}{2} \).
Problem 2.268medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sin{\left(3 x \right)}}{3} \).
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Differentiate \( \displaystyle f(x) = \ln{\left(5 x + 2 \right)} \).
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Differentiate \( \displaystyle f(x) = \frac{\tan{\left(5 x \right)}}{5} \).
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Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x + 1 \right)}}{5} \).
Problem 2.279medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x + 2 \right)}}{2} \).
Problem 2.280medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(2 x \right)} \right)}}{2} \).
Problem 2.285medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(5 x \right)} \right)} \).
Problem 2.287medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(4 x \right)}}{4} \).
Problem 2.291medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \sin{\left(x + 2 \right)} \).
Problem 2.300medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\ln{\left(x - 3 \right)} \right)} \).
Problem 2.301medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(2 x + 1 \right)}}{2} \).
Problem 2.302medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{e^{2 x + 2}}{2} \).
Problem 2.303medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \cos{\left(4 x \right)}}{4} \).
Problem 2.305medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\cos{\left(3 x - 3 \right)}}{3} \).
Problem 2.306medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(5 x + 2 \right)}}{5} \).
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Differentiate \( \displaystyle f(x) = 3 \sqrt{\left(x - 3\right)^{2} + 1} \).
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Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{asin}{\left(2 x - 1 \right)}}{2} \).
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Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(2 x \right)}}{2} \).
Problem 2.507medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{2 \operatorname{asin}{\left(5 x \right)}}{5} \).
Problem 2.512medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{2} \).
Problem 2.541medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{4} \).
Problem 2.557medium✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x - 1 \right)} \).
Problem 2.560medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \).
Problem 2.563medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x \right)} \).
Problem 2.566medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3} \).
Problem 2.574medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(x - 1 \right)} \).
Problem 2.598medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(3 x + 1\right)^{2} + 1}}{3} \).
Problem 2.629medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(5 x - 3\right)^{2} + 1}}{5} \).
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Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \).
Problem 2.680medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{\left(5 x - 3\right)^{2} + 1} \).
Problem 2.702medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \).
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Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \).
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Differentiate \( \displaystyle f(x) = 5 \operatorname{asin}{\left(x - 1 \right)} \).
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Differentiate \( \displaystyle f(x) = - \frac{\sqrt{4 \left(2 x + 1\right)^{2} + 1}}{4} \).
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Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(5 x \right)}}{5} \).
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Differentiate \( \displaystyle f(x) = 5 \sqrt{\left(x - 1\right)^{2} + 1} \).
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Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(5 x - 1\right)^{2} + 1}}{5} \).
Problem 2.999medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \sqrt{\left(x - 1\right)^{2} + 1} \).
Problem 2.1055medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{\left(5 x - 1\right)^{2} + 1}}{5} \).
Problem 2.1070medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \operatorname{asin}{\left(x \right)} \).
Problem 2.1071medium Video✓ Nihil obstat
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Problem 2.1077medium Video✓ Every equation proved
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Problem 2.1079medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \operatorname{asin}{\left(x \right)} \).
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Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \).
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Differentiate \( \displaystyle f(x) = 3 \sqrt{\left(x + 1\right)^{2} + 1} \).
Problem 2.1289medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{\left(5 x + 2\right)^{2} + 1}}{5} \).
Problem 2.1346medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(5 x + 2\right)^{2} + 1}}{5} \).
Problem 2.1373medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{\left(5 x + 1\right)^{2} + 1}}{5} \).
Problem 2.1415medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x + 1 \right)} \).
Problem 2.1678medium✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \).
Problem 2.3hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.4hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{x + 1} \right)} \).
Problem 2.8hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \).
Problem 2.9hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(\ln{\left(2 x \right)} - \frac{1}{2}\right) \).
Problem 2.13hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(4 x - 1\right) e^{4 x}}{2} \).
Problem 2.15hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.18hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(x \right)} \right)} \).
Problem 2.21hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\sec{\left(x \right)} \right)} \).
Problem 2.23hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
Problem 2.24hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(5 x \right)} \right)}}{5} \).
Problem 2.27hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
Problem 2.28hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8} \).
Problem 2.32hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
Problem 2.33hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} \).
Problem 2.34hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{20} \).
Problem 2.41hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
Problem 2.48hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{x}{2} + \frac{\ln{\left(2 x - 1 \right)}}{4} \).
Problem 2.49hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(2 x - 1 \right)} - 5 x - \frac{5 \ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.51hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(4 x \right)}\right) \).
Problem 2.54hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \ln{\left(\cos{\left(5 x \right)} \right)} \).
Problem 2.56hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)} \).
Problem 2.58hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.61hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\sin{\left(2 x \right)} \right)}}{2} \).
Problem 2.63hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{20} \).
Problem 2.69hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6} \).
Problem 2.71hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(2 x \right)} - 1\right) \).
Problem 2.72hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{12} \).
Problem 2.77hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\cot{\left(2 x + 2 \right)} + \csc{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.80hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x e^{2 x + 1} \).
Problem 2.81hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{2 x + 1} \).
Problem 2.89hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x e^{4 x + 1} \).
Problem 2.94hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(3 x + 2 \right)} + \csc{\left(3 x + 2 \right)} \right)}}{3} \).
Problem 2.99hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \).
Problem 2.104hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \).
Problem 2.109hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x \right)}}{10} \).
Problem 2.111hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x - 2 \right)}}{4} \).
Problem 2.116hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{5} \).
Problem 2.118hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x \right)}}{6} \).
Problem 2.119hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \).
Problem 2.122hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.126hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 1 \right)} + \csc{\left(4 x - 1 \right)} \right)}}{4} \).
Problem 2.129hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \left(x - 4\right) e^{x - 3} \).
Problem 2.131hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{4} \).
Problem 2.132hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{2 x} \cos{\left(2 x + \frac{\pi}{4} \right)}}{4} \).
Problem 2.133hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x \ln{\left(4 x - 3 \right)} + x + \frac{3 \ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.134hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(2 x + 2 \right)} - 2 x + 2 \ln{\left(x + 1 \right)} \).
Problem 2.136hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.139hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{12} \).
Problem 2.140hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{6} \).
Problem 2.144hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x + \frac{2}{5}\right) e^{5 x + 2} \).
Problem 2.148hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(5 x + 1\right) e^{5 x + 2} \).
Problem 2.149hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(3 x - 3 \right)} + \sec{\left(3 x - 3 \right)} \right)}}{3} \).
Problem 2.150hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 2\right) \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \).
Problem 2.154hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x + 2 \right)}}{2} \).
Problem 2.157hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(5 x \right)} - 1\right) \).
Problem 2.162hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(3 x \right)}\right) \).
Problem 2.164hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{2} - x\right) e^{2 x} \).
Problem 2.166hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} \).
Problem 2.173hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \).
Problem 2.175hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - 1\right) e^{2 x} \).
Problem 2.179hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x \right)}}{8} \).
Problem 2.181hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{2}\right) e^{2 x} \).
Problem 2.182hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5} \).
Problem 2.185hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.186hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
Problem 2.188hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(2 x \right)} \right)}}{2} \).
Problem 2.190hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\cot{\left(4 x - 3 \right)} + \csc{\left(4 x - 3 \right)} \right)}}{4} \).
Problem 2.191hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x - 2 \right)}}{20} \).
Problem 2.193hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{4}\right) e^{4 x} \).
Problem 2.196hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 3 \ln{\left(\cos{\left(x \right)} \right)} \).
Problem 2.197hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \).
Problem 2.203hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{2 \ln{\left(\cot{\left(3 x - 1 \right)} + \csc{\left(3 x - 1 \right)} \right)}}{3} \).
Problem 2.206hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.207hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.211hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{20}{3}\right) e^{3 x - 3} \).
Problem 2.213hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.218hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cot{\left(3 x + 1 \right)} + \csc{\left(3 x + 1 \right)} \right)}}{3} \).
Problem 2.223hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x - 6 \right)}}{20} \).
Problem 2.228hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x e^{x + 1} \).
Problem 2.231hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.234hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} \).
Problem 2.237hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x^{2} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} \).
Problem 2.238hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2} \).
Problem 2.251hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x + 2 \right)} - x + \frac{2 \ln{\left(3 x + 2 \right)}}{3} \).
Problem 2.254hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x - 3 \right)} - x - \ln{\left(x - 1 \right)} \).
Problem 2.255hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x + 1\right) e^{2 x + 2} \).
Problem 2.258hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(4 x \right)} \right)}}{4} \).
Problem 2.263hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(3 x \right)} - 1\right) \).
Problem 2.265hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{12} \).
Problem 2.270hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\cos{\left(5 x \right)} \right)}}{5} \).
Problem 2.272hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x^{2} \sec{\left(x \right)} \right)} \).
Problem 2.273hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{\ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
Problem 2.277hard Video✓ Checked independently
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\cos{\left(4 x \right)} \right)}}{4} \).
Problem 2.282hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(1 - x\right) e^{4 x - 3} \).
Problem 2.283hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{2 x} \cos{\left(2 x + \frac{\pi}{4} \right)}}{4} \).
Problem 2.288hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(10 \ln{\left(4 x \right)} - 5\right) \).
Problem 2.290hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{4 x + 1} \).
Problem 2.292hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x + \frac{5}{2}\right) e^{2 x + 2} \).
Problem 2.293hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \ln{\left(4 x - 3 \right)} - 5 x - \frac{15 \ln{\left(4 x - 3 \right)}}{4} \).
Problem 2.294hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(x + 2 \right)} - 3 x + 6 \ln{\left(x + 2 \right)} \).
Problem 2.296hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \).
Problem 2.298hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(4 x \right)} \right)}}{4} \).
Problem 2.304hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 5 \ln{\left(\cos{\left(x \right)} \right)} \).
Problem 2.309hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(3 x - 3 \right)}}{3 x - 2} \).
Problem 2.312hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x + 2 \right)}}{4 x + 2} \).
Problem 2.313hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\cot{\left(2 x + 1 \right)} + \csc{\left(2 x + 1 \right)} \right)}}{2} \).
Problem 2.318hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{2} \).
Problem 2.319hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x + \ln{\left(x^{2} \right)} + 6 \).
Problem 2.351hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4} \).
Problem 2.399hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x - 2 \right)}}{2} \).
Problem 2.455hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.467hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.469hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x + 2 \right)}}{4} \).
Problem 2.525hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)} \).
Problem 2.530hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{4 x^{2} - 1}}{2 x} \right)} \).
Problem 2.532hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\cot{\left(2 x - 1 \right)} \right)} \).
Problem 2.535hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(4 x^{2} e^{2 x} \right)} \).
Problem 2.536hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x \right)}}{2} \).
Problem 2.543hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{20} \).
Problem 2.550hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.552hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x \right)}}{8} \).
Problem 2.606hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.611hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x + 4 \right)}}{8} \).
Problem 2.619hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.673hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12} \).
Problem 2.684hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.691hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x \right)}}{8} \).
Problem 2.692hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x + 4 \right)}}{12} \).
Problem 2.696hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x \right)}}{4} \).
Problem 2.700hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x + 4 \right)}}{4} \).
Problem 2.718hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{2 x}}{2 x} \right)} \).
Problem 2.722hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(2 \sqrt{x^{2} + 1} \right)} \).
Problem 2.724hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \cos{\left(\frac{1}{x} \right)} \).
Problem 2.815hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x - 6 \right)}}{4} \).
Problem 2.822hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x \right)}}{4} \).
Problem 2.824hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x \right)}}{20} \).
Problem 2.856hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.857hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x + 2 \right)}}{20} \).
Problem 2.878hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x + 4 \right)}}{4} \).
Problem 2.888hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x - 6 \right)}}{4} \).
Problem 2.904hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(4 x^{2} e^{- 2 x} \right)} \).
Problem 2.907hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(2 x \right)} + \sec{\left(2 x \right)} \right)} \).
Problem 2.909hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\tan{\left(3 x \right)} + \sec{\left(3 x \right)} \right)} \).
Problem 2.910hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x + 2 \right)}}{12} \).
Problem 2.969hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x - 2 \right)}}{4} \).
Problem 2.974hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x + 2 \right)}}{4} \).
Problem 2.1017hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{- 3 x}}{9 x^{2}} \right)} \).
Problem 2.1031hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(2 x + 2 \sqrt{x \left(x - 1\right)} - 1 \right)} \).
Problem 2.1032hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\sqrt{1 - x^{2}}}{x} \right)} \).
Problem 2.1033hard✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x + 4 \right)}}{4} \).
Problem 2.1056hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x - 2 \right)}}{4} \).
Problem 2.1065hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \sin^{2}{\left(4 x - 1 \right)} + x \cos^{2}{\left(4 x - 1 \right)} - \frac{\sin{\left(8 x - 2 \right)}}{8} \).
Problem 2.1072hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x - 6 \right)}}{4} \).
Problem 2.1090hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(10 x - 6 \right)}}{20} \).
Problem 2.1098hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x + 2 \right)}}{6} \).
Problem 2.1100hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(10 x + 2 \right)}}{20} \).
Problem 2.1107hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x + 4 \right)}}{12} \).
Problem 2.1111hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x \right)}}{4} \).
Problem 2.1122hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{12} \).
Problem 2.1126hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{3 x}}{3 x} \right)} \).
Problem 2.1138hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{3 x}{\sqrt{1 - 9 x^{2}}} \right)} \).
Problem 2.1144hard✓ Every equation proved
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{\pi e^{- 3 x}}{4} \right)} \).
Problem 2.1145hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x - 2 \right)}}{6} \).
Problem 2.1168hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x - 6 \right)}}{6} \).
Problem 2.1170hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \).
Problem 2.1177hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x + 2 \right)}}{4} \).
Problem 2.1194hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(1 - x^{2} \right)} - \ln{\left(x^{2} + 1 \right)}}{2} \).
Problem 2.1201hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.1210hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x - 6 \right)}}{12} \).
Problem 2.1235hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x + 4 \right)}}{4} \).
Problem 2.1237hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \sin^{2}{\left(4 x + 2 \right)} + x \cos^{2}{\left(4 x + 2 \right)} - \frac{\sin{\left(8 x + 4 \right)}}{8} \).
Problem 2.1243hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x + 2 \right)}}{4} \).
Problem 2.1250hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12} \).
Problem 2.1258hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{- \ln{\left(x^{2} - 1 \right)} + \ln{\left(x^{2} + 1 \right)}}{2} \).
Problem 2.1262hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \).
Problem 2.1287hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(6 x + 4 \right)}}{6} \).
Problem 2.1294hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{8} \).
Problem 2.1299hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x - 6 \right)}}{2} \).
Problem 2.1313hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x - 6 \right)}}{4} \).
Problem 2.1336hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 6 \right)}}{12} \).
Problem 2.1356hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20} \).
Problem 2.1357hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x \right)}}{12} \).
Problem 2.1360hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{3 x}{\sqrt{9 x^{2} - 1}} \right)} \).
Problem 2.1363hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(2 x e^{2 x} \right)} \).
Problem 2.1364hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2} \).
Problem 2.1365hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(10 x + 4 \right)}}{20} \).
Problem 2.1368hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(10 x - 2 \right)}}{20} \).
Problem 2.1369hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x \right)}}{4} \).
Problem 2.1378hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x + 4 \right)}}{4} \).
Problem 2.1388hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{- 2 x}}{2 x} \right)} \).
Problem 2.1397hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x - 2 \right)}}{4} \).
Problem 2.1399hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x - 2 \right)}}{12} \).
Problem 2.1401hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(10 x + 4 \right)}}{20} \).
Problem 2.1403hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x + 4 \right)}}{4} \).
Problem 2.1423hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{3 x}{\sqrt{9 x^{2} + 1}} \right)} \).
Problem 2.1424hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x - 2 \right)}}{4} \).
Problem 2.1425hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x - 6 \right)}}{10} \).
Problem 2.1430hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x - 6 \right)}}{8} \).
Problem 2.1434hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x + 2 \right)}}{4} \).
Problem 2.1451hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(\frac{2 x}{\sqrt{4 x^{2} + 1}} \right)} \).
Problem 2.1453hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{3 x + 1}}{3 x + 1} \right)} \).
Problem 2.1454hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{1 - 2 x}}{\cos{\left(2 x - 1 \right)}} \right)} \).
Problem 2.1456hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x + \sqrt{x^{2} + 1} \right)} \).
Problem 2.1476hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{x^{2} - 2 x + 2} \).
Problem 2.1479hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{25 x^{2} + 10 x + 2}}{5} \).
Problem 2.1481hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{x^{2} - 6 x + 10} \).
Problem 2.1493hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{4 x^{2} + 4 x + 2}}{2} \).
Problem 2.1495hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{25 x^{2} - 30 x + 10}}{5} \).
Problem 2.1509hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{25 x^{2} - 30 x + 10}}{5} \).
Problem 2.1513hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{16 x^{2} + 8 x + 2}}{4} \).
Problem 2.1520hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{9 x^{2} + 6 x + 2} \).
Problem 2.1522hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{4 x^{2} + 8 x + 5}}{2} \).
Problem 2.1565hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{16 x^{2} + 16 x + 5}}{2} \).
Problem 2.1571hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{x^{2} + 4 x + 5} \).
Problem 2.1579hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{9 x^{2} - 18 x + 10}}{3} \).
Problem 2.1584hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x^{2} - 12 x + 10}}{2} \).
Problem 2.1608hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{16 x^{2} - 8 x + 2}}{4} \).
Problem 2.1656hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{4 x^{2} + 8 x + 5} \).
Problem 2.1658hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x^{2} - 4 x + 2}}{2} \).
Problem 2.1674hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{9 x^{2} - 6 x + 2} \).
Problem 2.1692hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \sqrt{x^{2} + 2 x + 2} \).
Problem 2.1738hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(10 x \right)}}{20} \).
Problem 2.1740hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x - 6 \right)}}{4} \).
Problem 2.1745hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 \sqrt{x^{2} - 6 x + 10} \).
Problem 2.1747hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{4 x^{2} - 4 x + 2}}{2} \).
Problem 2.1752hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{25 x^{2} + 10 x + 2} \).
Problem 2.1753hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{9 x^{2} - 18 x + 10}}{3} \).
Problem 2.1754hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sqrt{9 x^{2} + 6 x + 2}}{3} \).
Problem 2.1776hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{16 x^{2} - 8 x + 2}}{4} \).
Problem 2.1816hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x - 2 \right)}}{8} \).
Problem 2.1819hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \sqrt{x^{2} - 2 x + 2} \).
Problem 2.1831hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x + 4 \right)}}{2} \).
Problem 2.1833hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \sqrt{x^{2} + 4 x + 5} \).
Problem 2.1854hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{25 x^{2} - 10 x + 2} \).
Problem 2.1859hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(4 x + 2 \right)}}{8} \).
Problem 2.1891hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{2 \sqrt{25 x^{2} + 20 x + 5}}{5} \).
Problem 2.1892hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{9 x^{2} - 6 x + 2}}{3} \).
Problem 2.1912hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{25 x^{2} + 20 x + 5}}{5} \).
Problem 2.1918hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{4 x^{2} - 12 x + 10} \).
Problem 2.1936hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x - 6 \right)}}{4} \).
Problem 2.1976hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x^{2} + 4 x + 2}}{2} \).
Problem 2.1979hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(2 x + 2 \right)}}{4} \).
Problem 2.1998hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(10 x + 4 \right)}}{20} \).
Problem 2.1999hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{16 x^{2} + 16 x + 5}}{4} \).
Problem 2.2002hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x - 6 \right)}}{4} \).
Problem 2.2004hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{16 x^{2} - 24 x + 10}}{4} \).
Problem 2.2052hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{3 \sin{\left(2 x - 2 \right)}}{4} \).
Problem 2.2056hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(6 x + 2 \right)}}{12} \).
Problem 2.2067hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{4} \).
Problem 2.2070hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{\sin{\left(10 x + 2 \right)}}{4} \).
Problem 2.2081hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{25 x^{2} - 10 x + 2}}{5} \).
Problem 2.2082hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x + 4 \right)}}{10} \).
Problem 2.2137hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{16 x^{2} - 24 x + 10}}{2} \).
Problem 2.2145hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x \sin^{2}{\left(4 x - 3 \right)} + x \cos^{2}{\left(4 x - 3 \right)} - \frac{\sin{\left(8 x - 6 \right)}}{8} \).
Problem 2.2183hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x - 2 \right)}}{10} \).
Problem 2.2185hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \).
Problem 2.2191hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \).
Problem 2.2263hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{16 x^{2} + 8 x + 2}}{4} \).
Problem 2.2264hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x}{2} - \frac{5 \sin{\left(6 x - 2 \right)}}{12} \).
Problem 2.2271hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(10 x + 2 \right)}}{10} \).
Problem 2.2275hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x \sin^{2}{\left(4 x + 1 \right)} + x \cos^{2}{\left(4 x + 1 \right)} - \frac{\sin{\left(8 x + 2 \right)}}{8} \).
Problem 2.2279hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x + 4 \right)}}{4} \).
Problem 2.2379hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x}{2} - \frac{\sin{\left(6 x - 2 \right)}}{4} \).
Problem 2.2399hard Video✓ Nihil obstat