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Product rule practice problems

The derivative of a product: (fg)′ = f′g + fg′. 264 problems with worked solutions; in 263 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 2.1839

hard Beautiful Video
Differentiate \( \displaystyle f(x) = 5 x e^{x} - 5 e^{x} \).
  1. \[ \frac{d}{d x} \left(5 x e^{x} - 5 e^{x}\right) \]
    sumStart with the derivative of the entire expression.✓ Proved
  2. \[ = \frac{d}{d x} 5 x e^{x} - \frac{d}{d x} 5 e^{x} \]
    sumApply the sum rule to differentiate each term separately.✓ Proved
  3. \[ = 5 \frac{d}{d x} x e^{x} - 5 \frac{d}{d x} e^{x} \]
    constant-multiple algebraFactor out the constant 5 from both terms. Factor out the constant 5.✓ Proved
  4. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} \frac{d}{d x} x - 5 \frac{d}{d x} e^{x} \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} - 5 \frac{d}{d x} e^{x} \]
    derivative algebraDifferentiate x with respect to x. Distribute the derivative operator.✓ Proved
  6. \[ = 5 \left(x - 1\right) \frac{d}{d x} e^{x} + 5 e^{x} \]
    algebraFactor out the common derivative term.✓ Proved
  7. \[ = 5 \left(x - 1\right) e^{x} + 5 e^{x} \]
    exponentialDifferentiate the exponential function exp(x).✓ Proved
  8. \[ = 5 x e^{x} \]
    algebra simplifyDistribute the term (x - 1). Combine like terms to get the final result.✓ Proved
Answer \( 5 x e^{x} \)
Top: the graph of f = 5x·e^(x) - 5e^(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x·e^(x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(5 x e^{x} - 5 e^{x}\right) \]
    derivative✓ Proved
  2. \[ = 5 \frac{d}{d x} x e^{x} - 5 \frac{d}{d x} e^{x} \]
    product✓ Proved
  3. \[ = 5 x e^{x} \]
    chain sum 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 2.663

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{3 x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \).
  1. \[ \frac{d}{d x} \frac{3 x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{3 \frac{d}{d x} x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \]
    constant-multiplePull out the constant factor 3/4.✓ Proved
  3. \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \frac{3 \left(2 \ln{\left(x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]
    productApply the product rule.✓ Proved
  4. \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \frac{3 x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
    derivativeDifferentiate x**2.✓ Proved
  5. \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{3 x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
    derivativeDifferentiate the terms inside the parentheses.✓ Proved
  6. \[ = \frac{3 x \left(2 \ln{\left(x \right)} - 1\right)}{2} + \frac{3 x}{2} \]
    derivative algebraDifferentiate log(x). Simplify the term x**2 * (2/x).✓ Proved
  7. \[ = 3 x \ln{\left(x \right)} \]
    algebra simplify simplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved
Answer \( 3 x \log{\left(x \right)} \)
Top: the graph of f = 3x²·(2log(x) - 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x·(2log(x) - 1)/2 + 3x/2.
A second solution, written separately

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

  1. \[ \frac{d}{d x} \frac{3 x^{2} \left(2 \ln{\left(x \right)} - 1\right)}{4} \]
    derivative✓ Proved
  2. \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \left(2 \ln{\left(x \right)} - 1\right) \frac{d}{d x} \frac{3 x^{2}}{4} \]
    product✓ Proved
  3. \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(x \right)} - 1\right)}{4} + \frac{3 \left(2 \ln{\left(x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]
    constant-multiple✓ Proved
  4. \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{3 x \left(2 \ln{\left(x \right)} - 1\right)}{2} \]
    power✓ Proved
  5. \[ = \frac{3 x \left(2 \ln{\left(x \right)} - 1\right)}{2} + \frac{3 x}{2} \]
    logarithmic simplify simplify simplify simplify✓ Proved
  6. \[ = 3 x \left(\ln{\left(x \right)} - \frac{1}{2}\right) + \frac{3 x}{2} \]
    simplify✓ Proved
  7. \[ = 3 x \ln{\left(x \right)} \]
    simplify 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 2.1566

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \).
  1. \[ \frac{d}{d x} \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \frac{d}{d x} e^{x}}{2} + \frac{e^{x} \frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{2} \]
    productApply the product rule.✓ Proved
  4. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \frac{d}{d x} e^{x}}{2} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
    derivative algebraDifferentiate the first part of the product. Simplify the signs.✓ Proved
  5. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
    derivativeDifferentiate the second part of the product.✓ Proved
  6. \[ = e^{x} \sin{\left(x \right)} \]
    algebra simplify simplifyFactor out exp(x). Combine like terms. Final simplification.✓ Proved
Answer \( e^{x} \sin{\left(x \right)} \)
Top: the graph of f = (sin(x) - cos(x))·e^(x)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (sin(x) - cos(x))·e^(x)/2 + (sin(x) + cos(x))·e^(x)/2.
A second solution, written separately

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

  1. \[ \frac{d}{d x} \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
    derivative✓ Proved
  2. \[ = \frac{\frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} \]
    constant-multiple✓ Proved
  3. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \frac{d}{d x} e^{x}}{2} + \frac{e^{x} \frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)}{2} \]
    product✓ Proved
  4. \[ = \frac{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}}{2} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}}{2} \]
    derivative✓ Proved
  5. \[ = e^{x} \sin{\left(x \right)} \]
    algebra simplify 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 2.1595

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{3 x^{2} \ln{\left(x \right)}}{2} - \frac{3 x^{2}}{4} \).
  1. \[ \frac{d}{d x} \left(\frac{3 x^{2} \ln{\left(x \right)}}{2} - \frac{3 x^{2}}{4}\right) \]
    algebraDifferentiate the function with respect to x. Distribute the constants.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
    sumApply the sum rule for derivatives.✓ Proved
  3. \[ = - \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{3 \frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiplePull out the constant factors.✓ Proved
  4. \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{3 \ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} - \frac{3 \frac{d}{d x} x^{2}}{4} \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = 3 x \ln{\left(x \right)} \]
    derivative simplify constant-multiple simplifyDifferentiate the individual components. Simplify the expression inside the parentheses. Distribute the constant 3/2. Combine like terms.✓ Proved
Answer \( 3 x \ln{\left(x \right)} \)
Top: the graph of f = 3x²·log(x)/2 - 3x²/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x·log(x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{3 x^{2} \ln{\left(x \right)}}{2} - \frac{3 x^{2}}{4}\right) \]
    derivativeStart with the derivative of the whole function✓ Proved
  2. \[ = - \frac{d}{d x} \frac{3 x^{2}}{4} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
    sumSplit into two derivatives✓ Proved
  3. \[ = - 3 \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiplePull out constant 3 from second term✓ Proved
  4. \[ = - \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiplePull out 1/4 from derivative✓ Proved
  5. \[ = - \frac{3 x}{2} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]
    power simplifyDifferentiate x**2 Simplify the fraction✓ Proved
  6. \[ = - \frac{3 x}{2} + 3 \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiplePull out constant 3 from first term✓ Proved
  7. \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} - \frac{3 x}{2} + \frac{3 \ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} \]
    productApply product rule to x**2*log(x)✓ Proved
  8. \[ = 3 x \ln{\left(x \right)} \]
    power simplify simplify simplify simplify simplify simplifyDifferentiate x**2 and log(x) Simplify x**2*(1/x) to x Factor 3 inside the parentheses Distribute the 3 Combine like terms Cancel the 3*x terms Divide by 2 to get the 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 2.1768

hard Beautiful Video
Differentiate \( \displaystyle f(x) = \frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4} \).
  1. \[ \frac{d}{d x} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4}\right) \]
    sumStart with the derivative of the entire function.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
    sum constant-multipleApply the sum rule. Pull out the constant factors.✓ Proved
  3. \[ = - \frac{\frac{d}{d x} x^{2}}{4} + \frac{\frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
    constant-multipleSeparate the constants from the derivatives.✓ Proved
  4. \[ = \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \frac{\ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} - \frac{\frac{d}{d x} x^{2}}{4} \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = x \ln{\left(x \right)} \]
    derivative algebra algebra simplifyDifferentiate the individual components. Simplify the term x**2/x. Distribute the constant 1/2. Combine like terms.✓ Proved
Answer \( x \ln{\left(x \right)} \)
Top: the graph of f = x²·log(x)/2 - x²/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x·log(x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{x^{2} \ln{\left(x \right)}}{2} - \frac{x^{2}}{4}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
    sum✓ Proved
  3. \[ = - \frac{x}{2} + \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiple✓ Proved
  4. \[ = \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} - \frac{x}{2} + \frac{\ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} \]
    product✓ Proved
  5. \[ = x \ln{\left(x \right)} \]
    derivative simplify 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 2.1783

hard Beautiful Video
Differentiate \( \displaystyle f(x) = - \frac{x^{2} \ln{\left(x \right)}}{2} + \frac{x^{2}}{4} \).
  1. \[ \frac{d}{d x} \left(- \frac{x^{2} \ln{\left(x \right)}}{2} + \frac{x^{2}}{4}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2} \ln{\left(x \right)}}{2}\right) \]
    sum constant-multipleApply the sum rule. Pull out the constant factors.✓ Proved
  3. \[ = \frac{\frac{d}{d x} x^{2}}{4} - \frac{\frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
    constant-multipleSeparate the constants from the derivatives.✓ Proved
  4. \[ = - \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} - \frac{\ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} + \frac{\frac{d}{d x} x^{2}}{4} \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = - x \ln{\left(x \right)} \]
    derivative algebra algebra simplifyDifferentiate the individual components. Simplify the terms inside the parentheses and the second term. Distribute the constant -1/2. Combine like terms.✓ Proved
Answer \( - x \ln{\left(x \right)} \)
Top: the graph of f = -x²·log(x)/2 + x²/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -x·log(x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{x^{2} \ln{\left(x \right)}}{2} + \frac{x^{2}}{4}\right) \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} \frac{x^{2}}{4} - \frac{d}{d x} \frac{x^{2} \ln{\left(x \right)}}{2} \]
    sum✓ Proved
  3. \[ = \frac{d}{d x} \frac{x^{2}}{4} - \frac{\frac{d}{d x} x^{2} \ln{\left(x \right)}}{2} \]
    constant-multiple✓ Proved
  4. \[ = - \frac{x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} - \frac{\ln{\left(x \right)} \frac{d}{d x} x^{2}}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    product✓ Proved
  5. \[ = - x \ln{\left(x \right)} - \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    derivative simplify simplify✓ Proved
  6. \[ = - x \ln{\left(x \right)} - \frac{x}{2} + \frac{\frac{d}{d x} x^{2}}{4} \]
    constant-multiple✓ Proved
  7. \[ = - x \ln{\left(x \right)} \]
    derivative simplify rewrite simplifyRewrite -(2*x*log(x)+x)/2 as -x*log(x)-x/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 2.1923

hard Beautiful Video
Differentiate \( \displaystyle f(x) = x^{2} \ln{\left(x \right)} - \frac{x^{2}}{2} \).
  1. \[ \frac{d}{d x} \left(x^{2} \ln{\left(x \right)} - \frac{x^{2}}{2}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = - \frac{d}{d x} 0.5 x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    constantRewrite the fraction as a constant multiple.✓ Proved
  4. \[ = - 0.5 \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    constant-multiplePull out the constant factor.✓ Proved
  5. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2} - 0.5 \frac{d}{d x} x^{2} \]
    productApply the product rule to the first term.✓ Proved
  6. \[ = 2 x \ln{\left(x \right)} \]
    derivative algebra simplifyDifferentiate each part. Simplify the terms. Combine like terms.✓ Proved
Answer \( 2 x \ln{\left(x \right)} \)
Top: the graph of f = x²·log(x) - x²/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·log(x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(x^{2} \ln{\left(x \right)} - \frac{x^{2}}{2}\right) \]
    derivative✓ Proved
  2. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2} - \frac{d}{d x} \frac{x^{2}}{2} \]
    productApply product rule to x**2*log(x)✓ Proved
  3. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} + 2 x \ln{\left(x \right)} - \frac{d}{d x} \frac{x^{2}}{2} \]
    powerDerivative of x**2 is 2*x✓ Proved
  4. \[ = 2 x \ln{\left(x \right)} + x - \frac{d}{d x} \frac{x^{2}}{2} \]
    logarithmic simplifyDerivative of log(x) is 1/x Simplify x**2*(1/x) to x✓ Proved
  5. \[ = 2 x \ln{\left(x \right)} + x - \frac{\frac{d}{d x} x^{2}}{2} \]
    constant-multipleRewrite derivative of x**2/2✓ Proved
  6. \[ = 2 x \ln{\left(x \right)} \]
    power simplify simplifyDerivative of x**2 is 2*x Simplify 2*x/2 to x Cancel +x and -x✓ 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 2.705

hard Video
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4} \).
  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(2 x - 1 \right)}}{4} \]
    sumApply the sum rule.✓ Proved
  3. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]
    constantFactor out the constant from the last term.✓ Proved
  4. \[ = - x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - x \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(x - 1\right) - \left(x - 1\right) \ln{\left(2 x - 1 \right)} \frac{d}{d x} x - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]
    productApply the product rule to the second term.✓ Proved
  5. \[ = - x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]
    derivativeDifferentiate the individual parts of the product rule.✓ Proved
  6. \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{1}{2 \left(2 x - 1\right)} \]
    algebraSimplify the derivative of the log term.✓ Proved
  8. \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} + x - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{4 x - 2} \]
    derivativeEvaluate the remaining derivatives.✓ Proved
  9. \[ = x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{4 x - 2} - \frac{2 x^{2} - 2 x}{2 x - 1} \]
    algebra algebra algebraCombine the log terms. Group the rational terms.✓ Proved
  10. \[ = x - \frac{2 x}{4 x - 2} - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - 2 x}{2 x - 1} \]
    algebra algebraFind a common denominator for the rational part. Simplify the numerator of the rational part.✓ Proved
  11. \[ = x - \frac{x}{2 x - 1} - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - 2 x}{2 x - 1} \]
    algebraSimplify the fraction.✓ Proved
  12. \[ = x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - x}{2 x - 1} \]
    algebra algebraCombine the rational terms over the common denominator. Simplify the numerator.✓ Proved
  13. \[ = - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \]
    algebra algebraFactor out x from the numerator. Simplify the fraction.✓ Proved
  14. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} \]
    algebraFinal simplification.✓ Proved
Answer \( \left(1 - 2 x\right) \log{\left(2 x - 1 \right)} \)
Top: the graph of f = x²/2 - x·(x - 1)·log(2x - 1) - x/2 - log(2x - 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x·(x - 1)/(2x - 1) - x·log(2x - 1) + x - (x - 1)·log(2x - 1) - 1/2 - 1/(2(2x - 1)).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4}\right) \]
    derivative✓ Proved
  2. \[ = - x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} - \frac{1}{2} \]
    product✓ Proved
  3. \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} + x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{2 \left(2 x - 1\right)} \]
    chain✓ Proved
  4. \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} + x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{4 x - 2} \]
    simplify algebra✓ Proved
  5. \[ = x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{4 x \left(x - 1\right) + 1}{4 x - 2} \]
    simplifyCombined rational 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 2.2440

hard Video
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)}\right) \]
    derivativeDifferentiate the entire function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \]
    sum constant-multipleApply the sum rule. Factor out the constants.✓ Proved
  3. \[ = \frac{\frac{d}{d x} x}{2} - \frac{3 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \]
    constant-multipleFactor out the constants from each term.✓ Proved
  4. \[ = - \frac{3 x}{2} + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    powerDifferentiate the power terms.✓ Proved
  5. \[ = - \frac{3 x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) + \frac{1}{2} \]
    productApply the product rule to the third term.✓ Proved
  6. \[ = - \frac{3 x}{2} + \left(- \frac{d}{d x} x + \frac{3 \frac{d}{d x} x^{2}}{2}\right) \ln{\left(3 x - 1 \right)} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    sumDifferentiate the polynomial inside the product rule.✓ Proved
  7. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    power algebraDifferentiate the power terms in the first part of the product rule. Simplify the expression inside the parentheses.✓ Proved
  8. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{\left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} \]
    chainApply the chain rule to the logarithm.✓ Proved
  9. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{3 \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right)}{3 x - 1} \]
    constant-multipleDifferentiate the inner function 3x - 1.✓ Proved
  10. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{\frac{9 x^{2}}{2} - 3 x + \frac{1}{2}}{3 x - 1} \]
    algebra simplify algebraMultiply the 3 into the numerator. Simplify the first two terms. Factor out 1/2 from the numerator.✓ Proved
  11. \[ = \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \]
    algebra simplify algebra simplifyRecognize the perfect square trinomial. Cancel the common factor (3x - 1). Distribute the 1/2. Combine all terms to find the final simplified derivative.✓ Proved
Answer \( \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} \)
Top: the graph of f = -3x²/4 + x/2 + (3x²/2 - x + 1/6)·log(3x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x/2 + (3x - 1)·log(3x - 1) + 1/2 + 3(3x²/2 - x + 1/6)/(3x - 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{4} + \frac{x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{3 x}{2} + \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} \]
    sum✓ Proved
  3. \[ = - \frac{3 x}{2} + \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right) + \frac{1}{2} \]
    product✓ Proved
  4. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{3 \left(\frac{3 x^{2}}{2} - x + \frac{1}{6}\right)}{3 x - 1} \]
    derivative simplify✓ Proved
  5. \[ = - \frac{3 x}{2} + \left(3 x - 1\right) \ln{\left(3 x - 1 \right)} + \frac{1}{2} + \frac{\frac{9 x^{2}}{2} - 3 x + \frac{1}{2}}{3 x - 1} \]
    simplifyCombined the fraction✓ 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 2.1779

hard Video
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{4} + \frac{x}{2} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \).
  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} + \frac{x}{2} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)}\right) \]
    sumStart with the derivative of the entire expression.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \]
    sumApply the sum rule to separate the terms.✓ Proved
  3. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    productApply the product rule to the third term.✓ Proved
  4. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{2} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    sumApply the sum rule to the derivative of the polynomial.✓ Proved
  5. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    constantThe derivative of the constant 1/2 is 0.✓ Proved
  6. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} x - \ln{\left(x + 1 \right)} \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    algebra algebraDistribute the log(x + 1) term. Remove the zero term.✓ Proved
  7. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    algebraFactor out log(x + 1) from the parenthesis.✓ Proved
  8. \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
    logarithmicDifferentiate the log(x + 1) term.✓ Proved
  9. \[ = \frac{x}{2} - \left(x + 1\right) \ln{\left(x + 1 \right)} + \frac{1}{2} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
    power algebra algebraDifferentiate the power terms. Simplify the coefficients. Factor out 1/2 from the numerator.✓ Proved
  10. \[ = - \left(x + 1\right) \ln{\left(x + 1 \right)} \]
    algebra algebra algebraRecognize the perfect square trinomial. Cancel the common factor (x + 1). Distribute the division by 2.✓ Proved
  11. \[ = \left(- x - 1\right) \ln{\left(x + 1 \right)} \]
    simplifyCombine all remaining terms.✓ Proved
Answer \( \left(- x - 1\right) \ln{\left(x + 1 \right)} \)
Top: the graph of f = x²/4 + x/2 - (x²/2 + x + 1/2)·log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x/2 - (x + 1)·log(x + 1) + 1/2 - (x²/2 + x + 1/2)/(x + 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} + \frac{x}{2} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)}\right) \]
    derivativeStart with the derivative of the whole function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \]
    sumSeparate the sum of terms.✓ Proved
  3. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \ln{\left(x + 1 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    productApply product rule to the last term.✓ Proved
  4. \[ = - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{2} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    sumExpand the derivative of the sum inside the product.✓ Proved
  5. \[ = - \left(\frac{d}{d x} \frac{x^{2}}{2} + 1\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    constant simplifyDerivative of constants is zero. Combine constants.✓ Proved
  6. \[ = - \left(x \frac{d}{d x} x + 1\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    powerDifferentiate x**2/2 using power rule.✓ Proved
  7. \[ = - \left(x + 1\right) \ln{\left(x + 1 \right)} - \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    derivative simplifyDerivative of x is 1. Combine terms.✓ Proved
  8. \[ = - \left(x + 1\right) \ln{\left(x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{4} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
    logarithmic simplifyDerivative of log(x+1). Rewrite the fraction.✓ Proved
  9. \[ = \frac{x}{2} - \left(x + 1\right) \ln{\left(x + 1 \right)} + \frac{1}{2} - \frac{\frac{x^{2}}{2} + x + \frac{1}{2}}{x + 1} \]
    power simplifyDifferentiate x**2/4 and x/2. Distribute the minus sign.✓ Proved
  10. \[ = - \left(x + 1\right) \ln{\left(x + 1 \right)} \]
    simplifySimplify the fraction: (x**2/2 + x + 1/2)/(x + 1) = (x + 1)/2.✓ Proved
  11. \[ = \left(- x - 1\right) \ln{\left(x + 1 \right)} \]
    simplifyCombine the remaining terms: x/2 + 1/2 - (x + 1)/2 = 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 2.2182

hard Video
Differentiate \( \displaystyle f(x) = - x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \).
  1. \[ \frac{d}{d x} \left(- x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - 2 x + \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} \]
    sumDifferentiate the terms separately.✓ Proved
  3. \[ = - 2 x + \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) - \frac{1}{2} \]
    productApply the product rule to the second term.✓ Proved
  4. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} + \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} - \frac{1}{2} \]
    derivativeDifferentiate the polynomial factor.✓ Proved
  5. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{\left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} \]
    chainApply the chain rule to the logarithm.✓ Proved
  6. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{4 \left(2 x^{2} + x + \frac{1}{8}\right)}{4 x + 1} \]
    derivativeDifferentiate the inner function of the logarithm.✓ Proved
  7. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{8 x^{2} + 4 x + \frac{1}{2}}{4 x + 1} \]
    algebra algebra algebra algebra algebra simplify simplifySimplify the expression. Distribute the 4 into the numerator. Add and subtract 1/2 to facilitate simplification. Combine terms over a common denominator. Expand the numerator. Simplify the fraction. Remove zero.✓ Proved
  8. \[ = \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \]
    algebra algebra simplifyPerform polynomial division on the fraction. Combine the linear terms. Final simplification.✓ Proved
Answer \( \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \)
Top: the graph of f = -x² - x/2 + (2x² + x + 1/8)·log(4x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x + (4x + 1)·log(4x + 1) - 1/2 + 4(2x² + x + 1/8)/(4x + 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- x^{2} - \frac{x}{2} + \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} x^{2} + \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \]
    sum✓ Proved
  3. \[ = - 2 x + \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} \]
    power✓ Proved
  4. \[ = - 2 x + \left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \ln{\left(4 x + 1 \right)} + \ln{\left(4 x + 1 \right)} \frac{d}{d x} \left(2 x^{2} + x + \frac{1}{8}\right) - \frac{1}{2} \]
    product✓ Proved
  5. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{\left(2 x^{2} + x + \frac{1}{8}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} \]
    product✓ Proved
  6. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{4 \left(2 x^{2} + x + \frac{1}{8}\right)}{4 x + 1} \]
    chain simplify algebra✓ Proved
  7. \[ = - 2 x + \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} - \frac{1}{2} + \frac{8 x^{2} + 4 x + \frac{1}{2}}{4 x + 1} \]
    simplify✓ Proved
  8. \[ = \left(4 x + 1\right) \ln{\left(4 x + 1 \right)} \]
    simplifyFraction simplifies to 2*x + 1/2, canceling with -2*x - 1/2✓ 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 2.2461

hard Video
Differentiate \( \displaystyle f(x) = - \frac{9 x^{2}}{4} - 3 x + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{9 x^{2}}{4} - 3 x + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} \left(- \frac{9 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \]
    constant-multiplePull out the constant -9/4.✓ Proved
  4. \[ = \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
    productApply the product rule to the third term.✓ Proved
  5. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
    derivative algebraDifferentiate the polynomial inside the product. Simplify the polynomial derivative.✓ Proved
  6. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{\left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{3 \left(\frac{9 x^{2}}{2} + 6 x + 2\right)}{3 x + 2} \]
    derivativeDifferentiate the inner function 3*x + 2.✓ Proved
  8. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{\frac{27 x^{2}}{2} + 18 x + 6}{3 x + 2} \]
    algebraDistribute the 3 into the numerator.✓ Proved
  9. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{27 x^{2} + 36 x + 12}{6 x + 4} \]
    algebra algebraFind a common denominator for the fraction term. Factor out 3 from the numerator.✓ Proved
  10. \[ = \frac{3 \left(3 x + 2\right)^{2}}{6 x + 4} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} \]
    algebraRecognize the perfect square trinomial.✓ Proved
  11. \[ = \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 \frac{d}{d x} x - \frac{9 \frac{d}{d x} x^{2}}{4} + 3 \]
    algebraCancel the common factor (3*x + 2).✓ Proved
  12. \[ = \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} \]
    derivative algebra simplifyEvaluate the remaining derivatives. Distribute and expand the terms. Combine like terms to get the final result.✓ Proved
Answer \( 3 \left(3 x + 2\right) \ln{\left(3 x + 2 \right)} \)
Top: the graph of f = -9x²/4 - 3x + (9x²/2 + 6x + 2)·log(3x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -9x/2 + (9x + 6)·log(3x + 2) - 3 + 3(9x²/2 + 6x + 2)/(3x + 2).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{9 x^{2}}{4} - 3 x + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} 3 x - \frac{d}{d x} \frac{9 x^{2}}{4} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} \]
    sum✓ Proved
  3. \[ = - \frac{\frac{d}{d x} 9 x^{2}}{4} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} - 3 \]
    constant-multiple✓ Proved
  4. \[ = - \frac{9 \frac{d}{d x} x^{2}}{4} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} - 3 \]
    constant-multiple✓ Proved
  5. \[ = - \frac{9 x}{2} + \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \ln{\left(3 x + 2 \right)} - 3 \]
    power simplify✓ Proved
  6. \[ = - \frac{9 x}{2} + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(\frac{9 x^{2}}{2} + 6 x + 2\right) - 3 \]
    product✓ Proved
  7. \[ = - \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} + \left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - 3 \]
    derivative✓ Proved
  8. \[ = - \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 + \frac{\left(\frac{9 x^{2}}{2} + 6 x + 2\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} \]
    chain✓ Proved
  9. \[ = - \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 + \frac{3 \left(\frac{9 x^{2}}{2} + 6 x + 2\right)}{3 x + 2} \]
    derivative simplify✓ Proved
  10. \[ = - \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 + \frac{\frac{27 x^{2}}{2} + 18 x + 6}{3 x + 2} \]
    simplify✓ Proved
  11. \[ = - \frac{9 x}{2} + \left(9 x + 6\right) \ln{\left(3 x + 2 \right)} - 3 + \frac{27 x^{2} + 36 x + 12}{6 x + 4} \]
    simplify 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 2.654

hard Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \).
  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{5 x}\right)}{10} \]
    productApply the product rule.✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]
    derivativeDifferentiate the exponential term.✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- 5 e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + 5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]
    chain algebraApply the chain rule to the cosine term. Distribute the negative sign inside the parentheses.✓ Proved
  6. \[ = - \frac{\sqrt{2} \left(- e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{2} \]
    constant-multiple simplifyFactor out the common constant 5. Simplify the coefficient.✓ Proved
  7. \[ = - \frac{\sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)}\right) e^{5 x}}{2} \]
    algebraFactor out the exponential term.✓ Proved
  8. \[ = e^{5 x} \sin{\left(5 x \right)} \]
    algebra algebra algebra simplify algebra simplifyExpand the cosine and sine of the sum using angle addition identities. Substitute cos(pi/4) = sin(pi/4) = sqrt(2)/2. Distribute the negative sign through the second parentheses. Combine like terms inside the parentheses. Multiply the constants and handle the sign. Simplify the final expression.✓ Proved
Answer \( e^{5 x} \sin{\left(5 x \right)} \)
Top: the graph of f = -sqrt(2)·e^(5x)·cos(5x + pi/4)/10. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = sqrt(2)·e^(5x)·sin(5x + pi/4)/2 - sqrt(2)·e^(5x)·cos(5x + pi/4)/2.
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10}\right) \]
    derivativeStart with the derivative✓ Proved
  2. \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{10} \]
    constant-multiplePull out the constant✓ Proved
  3. \[ = - \frac{\sqrt{2} \left(e^{5 x} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{5 x}\right)}{10} \]
    productApply product rule✓ Proved
  4. \[ = - \frac{\sqrt{2} \left(- 5 e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)} + 5 e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}\right)}{10} \]
    derivative algebraDifferentiate each factor Simplify the minus sign✓ Proved
  5. \[ = - \frac{\sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} \right)} + \cos{\left(5 x + \frac{\pi}{4} \right)}\right) e^{5 x}}{2} \]
    algebra constant-multiple simplifyFactor out common terms Combine constants Reduce the fraction✓ Proved
  6. \[ = \frac{\sqrt{2} e^{5 x} \sin{\left(5 x + \frac{\pi}{4} \right)}}{2} - \frac{\sqrt{2} e^{5 x} \cos{\left(5 x + \frac{\pi}{4} \right)}}{2} \]
    algebraDistribute the factor✓ Proved
  7. \[ = \frac{\sqrt{2} \left(\sin{\left(5 x + \frac{\pi}{4} \right)} - \cos{\left(5 x + \frac{\pi}{4} \right)}\right) e^{5 x}}{2} \]
    simplifyReorder 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 2.1660

hard Video
Differentiate \( \displaystyle f(x) = - x^{2} + \frac{3 x}{2} + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \).
  1. \[ \frac{d}{d x} \left(- x^{2} + \frac{3 x}{2} + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
    productApply the product rule to the third term.✓ Proved
  4. \[ = \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) \]
    power algebraDifferentiate the polynomial part. Simplify the polynomial derivative.✓ Proved
  5. \[ = \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{\left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]
    chainApply the chain rule to the logarithm.✓ Proved
  6. \[ = \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{4 \left(2 x^{2} - 3 x + \frac{9}{8}\right)}{4 x - 3} \]
    derivativeDifferentiate the inner function of the logarithm.✓ Proved
  7. \[ = \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \left(- x^{2}\right) + \frac{8 x^{2} - 12 x + \frac{9}{2}}{4 x - 3} \]
    algebraMultiply the constant 4 into the numerator.✓ Proved
  8. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} + \frac{8 x^{2} - 12 x + \frac{9}{2}}{4 x - 3} \]
    derivativeEvaluate the remaining derivatives.✓ Proved
  9. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} + \frac{16 x^{2} - 24 x + 9}{8 x - 6} \]
    algebra algebraSimplify the fraction by multiplying numerator and denominator by 2. Distribute the 2 in the denominator.✓ Proved
  10. \[ = - 2 x + \frac{\left(4 x - 3\right)^{2}}{8 x - 6} + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} \]
    algebraNotice that the numerator is a perfect square.✓ Proved
  11. \[ = \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} \]
    simplify algebra simplifySimplify the fraction by canceling common terms. Combine the rational terms. The linear terms cancel out.✓ Proved
Answer \( \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} \)
Top: the graph of f = -x² + 3x/2 + (2x² - 3x + 9/8)·log(4x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x + (4x - 3)·log(4x - 3) + 3/2 + 4(2x² - 3x + 9/8)/(4x - 3).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- x^{2} + \frac{3 x}{2} + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)}\right) \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} \frac{3 x}{2} - \frac{d}{d x} x^{2} + \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \]
    sum✓ Proved
  3. \[ = - \frac{d}{d x} x^{2} + \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} \]
    constant-multiple✓ Proved
  4. \[ = - 2 x + \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} \]
    power✓ Proved
  5. \[ = - 2 x + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \ln{\left(4 x - 3 \right)} \frac{d}{d x} \left(2 x^{2} - 3 x + \frac{9}{8}\right) + \frac{3}{2} \]
    product✓ Proved
  6. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \ln{\left(4 x - 3 \right)} + \frac{3}{2} \]
    derivative✓ Proved
  7. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} + \frac{\left(2 x^{2} - 3 x + \frac{9}{8}\right) \frac{d}{d x} \left(4 x - 3\right)}{4 x - 3} \]
    chain✓ Proved
  8. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} + \frac{4 \left(2 x^{2} - 3 x + \frac{9}{8}\right)}{4 x - 3} \]
    derivative simplify✓ Proved
  9. \[ = - 2 x + \left(4 x - 3\right) \ln{\left(4 x - 3 \right)} + \frac{3}{2} + \frac{8 x^{2} - 12 x + \frac{9}{2}}{4 x - 3} \]
    simplifyfinal 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 2.1743

hard Video
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - \frac{x}{2} + \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \).
  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - \frac{x}{2} + \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)}\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
    productApply the product rule to the third term.✓ Proved
  4. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} + \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} \]
    derivativeDifferentiate the polynomial part.✓ Proved
  5. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{\left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
    chainApply the chain rule to the logarithm.✓ Proved
  6. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{2 \left(- x^{2} + x - \frac{1}{4}\right)}{2 x - 1} \]
    derivativeDifferentiate the inner function 2x - 1.✓ Proved
  7. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{- 2 x^{2} + 2 x - \frac{1}{2}}{2 x - 1} \]
    algebraSimplify the fraction.✓ Proved
  8. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{- 2 x^{2} + 2 x - \frac{1}{2}}{2 x - 1} \]
    derivative algebraDifferentiate the first two terms. Distribute the 2 in the numerator.✓ Proved
  9. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{- 4 x^{2} + 4 x - 1}{4 x - 2} \]
    algebraMultiply numerator and denominator by 2 to clear the fraction.✓ Proved
  10. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{4 x^{2} - 4 x + 1}{4 x - 2} \]
    algebraFactor out the negative sign.✓ Proved
  11. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{\left(2 x - 1\right)^{2}}{4 x - 2} - \frac{1}{2} \]
    algebraRecognize the perfect square trinomial.✓ Proved
  12. \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} \]
    algebra algebra simplifyCancel the common factor (2*x - 1). Split the fraction. Combine all terms to get the final answer.✓ Proved
Answer \( \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} \)
Top: the graph of f = x²/2 - x/2 + (-x² + x - 1/4)·log(2x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x + (1 - 2x)·log(2x - 1) - 1/2 + 2(-x² + x - 1/4)/(2x - 1).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - \frac{x}{2} + \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)}\right) \]
    derivativeStart with the derivative of the whole function✓ Proved
  2. \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} + \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \]
    sumSplit the derivative into three parts✓ Proved
  3. \[ = x + \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} \]
    derivativeDifferentiate the simple terms✓ Proved
  4. \[ = x + \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(- x^{2} + x - \frac{1}{4}\right) - \frac{1}{2} \]
    productApply product rule to the remaining product✓ Proved
  5. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} + \left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - \frac{1}{2} \]
    derivativeDifferentiate the first factor✓ Proved
  6. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{\left(- x^{2} + x - \frac{1}{4}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
    chainDifferentiate the logarithm✓ Proved
  7. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{2 \left(- x^{2} + x - \frac{1}{4}\right)}{2 x - 1} \]
    derivative simplifyDifferentiate the inner linear function Simplify the constant factor✓ Proved
  8. \[ = x + \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} + \frac{- 2 x^{2} + 2 x - \frac{1}{2}}{2 x - 1} \]
    simplifyCombine the fraction into a single rational expression✓ 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 2.2057

hard Video
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + \ln{\left(\left(2 x\right)^{x^{2}} \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + \ln{\left(\left(2 x\right)^{x^{2}} \right)}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \ln{\left(\left(2 x\right)^{x^{2}} \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \ln{\left(2 x \right)} \]
    rewriteRewrite the logarithm using the property log(a^b) = b*log(a).✓ Proved
  4. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right) \]
    algebraExpand the logarithm of the product.✓ Proved
  5. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} x^{2} \ln{\left(2 \right)} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    sumApply the sum rule again.✓ Proved
  6. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \ln{\left(2 \right)} \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    constant-multiplePull out the constant log(2).✓ Proved
  7. \[ = - x + \ln{\left(2 \right)} \frac{d}{d x} x^{2} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    derivativeDifferentiate the first term.✓ Proved
  8. \[ = - x + 2 x \ln{\left(2 \right)} + \frac{d}{d x} x^{2} \ln{\left(x \right)} \]
    derivativeDifferentiate the second term.✓ Proved
  9. \[ = x^{2} \frac{d}{d x} \ln{\left(x \right)} - x + 2 x \ln{\left(2 \right)} + \ln{\left(x \right)} \frac{d}{d x} x^{2} \]
    productApply the product rule to the third term.✓ Proved
  10. \[ = 2 x \ln{\left(x \right)} + 2 x \ln{\left(2 \right)} \]
    derivative algebra algebra simplifyDifferentiate the components of the product rule. Simplify the term x**2/x. Distribute the terms. Combine like terms.✓ Proved
  11. \[ = 2 x \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right) \]
    simplifyFactor out 2*x.✓ Proved
  12. \[ = 2 x \ln{\left(2 x \right)} \]
    simplifyCombine the logarithms.✓ Proved
Answer \( 2 x \ln{\left(2 x \right)} \)
Top: the graph of f = -x²/2 + log((2x)^(x²)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·log(2x).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + \ln{\left(\left(2 x\right)^{x^{2}} \right)}\right) \]
    derivative✓ Proved
  2. \[ = \frac{d}{d x} \left(- \frac{x^{2}}{2} + \ln{\left(e^{x^{2} \ln{\left(2 x \right)}} \right)}\right) \]
    rewrite✓ Proved
  3. \[ = \frac{d}{d x} \left(x^{2} \ln{\left(2 x \right)} - \frac{x^{2}}{2}\right) \]
    simplify✓ Proved
  4. \[ = - x + \frac{d}{d x} x^{2} \ln{\left(2 x \right)} \]
    derivative✓ Proved
  5. \[ = x^{2} \frac{d}{d x} \ln{\left(2 x \right)} + 2 x \ln{\left(2 x \right)} - x \]
    product✓ Proved
  6. \[ = 2 x \ln{\left(2 x \right)} \]
    derivative 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 2.2178

hard Video
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{4} - \frac{3 x}{2} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)} \).
  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} - \frac{3 x}{2} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)} \]
    sumApply the sum rule.✓ Proved
  3. \[ = \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    productApply the product rule to the third term.✓ Proved
  4. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} \]
    derivativeDifferentiate the polynomial part of the product.✓ Proved
  5. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{- \frac{x^{2}}{2} + 3 x - \frac{9}{2}}{x - 3} \]
    logarithmicDifferentiate the natural logarithm.✓ Proved
  6. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- \frac{x^{2}}{2} + 3 x - \frac{9}{2}}{x - 3} \]
    derivative simplifyDifferentiate the remaining power and constant terms. Simplify the first term.✓ Proved
  7. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- 0.5 x^{2} + 3 x - 4.5}{x - 3} \]
    algebraRewrite coefficients as decimals for clarity.✓ Proved
  8. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- 0.5 x^{2} + 3.0 x - 4.5}{x - 3} \]
    algebraFactor out -0.5 from the numerator.✓ Proved
  9. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} \]
    algebra simplify algebra algebra simplifyFactor the quadratic expression. Cancel the common factor (x - 3). Distribute the -0.5. Rewrite 0.5*x and 1.5 as fractions. Combine all terms to find the final simplified derivative.✓ Proved
Answer \( \left(3 - x\right) \ln{\left(x - 3 \right)} \)
Top: the graph of f = x²/4 - 3x/2 + (-x²/2 + 3x - 9/2)·log(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x/2 + (3 - x)·log(x - 3) - 3/2 + (-x²/2 + 3x - 9/2)/(x - 3).
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(\frac{x^{2}}{4} - \frac{3 x}{2} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)}\right) \]
    derivative✓ Proved
  2. \[ = - \frac{d}{d x} \frac{3 x}{2} + \frac{d}{d x} \frac{x^{2}}{4} + \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)} \]
    sum✓ Proved
  3. \[ = \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \ln{\left(x - 3 \right)} - \frac{3}{2} \]
    derivative✓ Proved
  4. \[ = \frac{x}{2} + \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) \frac{d}{d x} \ln{\left(x - 3 \right)} + \ln{\left(x - 3 \right)} \frac{d}{d x} \left(- \frac{x^{2}}{2} + 3 x - \frac{9}{2}\right) - \frac{3}{2} \]
    product✓ Proved
  5. \[ = \frac{x}{2} + \left(3 - x\right) \ln{\left(x - 3 \right)} - \frac{3}{2} + \frac{- \frac{x^{2}}{2} + 3 x - \frac{9}{2}}{x - 3} \]
    derivativeCorrected derivative of the product✓ Proved
  6. \[ = \left(3 - x\right) \ln{\left(x - 3 \right)} \]
    simplify simplify simplifySimplify the fraction by polynomial division Combine like 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 2.342

hard Video
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \).
  1. \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2}\right) \]
    Start with the derivative of the function.✓ Proved
  2. \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \]
    constant-multiplePull out the constant factor.✓ Proved
  3. \[ = - \frac{3 \sqrt{2} \left(e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 1}\right)}{2} \]
    productApply the product rule.✓ Proved
  4. \[ = - \frac{3 \sqrt{2} \left(e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeDifferentiate exp(x - 1).✓ Proved
  5. \[ = - \frac{3 \sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
    trigDifferentiate cos(x - 1 + pi/4).✓ Proved
  6. \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
    algebraFactor out exp(x - 1).✓ Proved
  7. \[ = - \frac{3 \sqrt{2} \left(- \cos{\left(- x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
    algebraUse the identity sin(theta) = cos(pi/2 - theta) or similar, but let's simplify the expression directly.✓ Proved
  8. \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
    simplifyThe expression is already in a standard form, but let's combine terms if possible. Actually, let's use the identity cos(a) - sin(a) = sqrt(2)cos(a + pi/4).✓ Proved
  9. \[ = 3 e^{x - 1} \sin{\left(x - 1 \right)} \]
    algebra algebra simplify trig simplifyUse the identity cos(u) - sin(u) = sqrt(2)cos(u + pi/4). Simplify the argument inside the cosine. Multiply the constants: (sqrt(2)*sqrt(2))/2 = 1. Use the identity cos(u + pi/2) = -sin(u). Simplify the final expression.✓ Proved
Answer \( 3 e^{x - 1} \sin{\left(x - 1 \right)} \)
Top: the graph of f = -3sqrt(2)·e^(x - 1)·cos(x - 1 + pi/4)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3sqrt(2)·e^(x - 1)·sin(x - 1 + pi/4)/2 - 3sqrt(2)·e^(x - 1)·cos(x - 1 + pi/4)/2.
A second solution, written separately

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

  1. \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2}\right) \]
    derivativeStart with the derivative of the given function.Not checked
  2. \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \]
    constant-multipleFactor out the constant coefficient -3*sqrt(2)/2.✓ Proved
  3. \[ = - \frac{3 \sqrt{2} \left(e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 1}\right)}{2} \]
    productApply the product rule to exp(x-1) and cos(x-1+pi/4).✓ Proved
  4. \[ = - \frac{3 \sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 1 + \frac{\pi}{4}\right) + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 1\right)\right)}{2} \]
    chainApply the chain rule to the exponential and cosine terms.✓ Proved
  5. \[ = - \frac{3 \sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]
    derivativeEvaluate the derivatives of the inner functions x-1 and x-1+pi/4, which are both 1.✓ Proved
  6. \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]
    algebraFactor out the common term exp(x-1).✓ Proved
  7. \[ = 3 e^{x - 1} \sin{\left(x - 1 \right)} \]
    trig simplifySimplify the trigonometric expression cos(A-pi/4)-sin(A-pi/4) where A=x-1 using angle sum identities. Multiply the constants: (-3*sqrt(2)/2) * (-sqrt(2)) = 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 2.387

hard Video
Differentiate \( \displaystyle f(x) = x^{2} \left(2 \ln{\left(2 x \right)} - 1\right) \).
  1. \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(2 x \right)} - 1\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + \left(2 \ln{\left(2 x \right)} - 1\right) \frac{d}{d x} x^{2} \]
    productApply the product rule.✓ Proved
  3. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
    powerDifferentiate the first part of the product.✓ Proved
  4. \[ = x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(2 x \right)}\right) + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
    sumApply the sum rule to the second part.✓ Proved
  5. \[ = 2 x^{2} \frac{d}{d x} \ln{\left(2 x \right)} + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
    constant constant-multipleDifferentiate the constant term. Factor out the constant 2.✓ Proved
  6. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + x \frac{d}{d x} 2 x \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \]
    derivative algebra algebraDifferentiate the inner function 2*x. Simplify the expression inside the parentheses. Simplify the second term.✓ Proved
  8. \[ = 4 x \ln{\left(2 x \right)} \]
    algebra algebra algebraFactor out 2*x. Simplify the expression inside the parentheses. Final simplification.✓ Proved
Answer \( 4 x \log{\left(2 x \right)} \)
Top: the graph of f = x²·(2log(2x) - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·(2log(2x) - 1) + 2x.
A second solution, written separately

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

  1. \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(2 x \right)} - 1\right) \]
    derivative✓ Proved
  2. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + \left(2 \ln{\left(2 x \right)} - 1\right) \frac{d}{d x} x^{2} \]
    product✓ Proved
  3. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
    derivative✓ Proved
  4. \[ = 2 x^{2} \frac{d}{d x} \ln{\left(2 x \right)} + 2 x \left(2 \ln{\left(2 x \right)} - 1\right) \]
    constant-multiple✓ Proved
  5. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + x \frac{d}{d x} 2 x \]
    chain✓ Proved
  6. \[ = 2 x \left(2 \ln{\left(2 x \right)} - 1\right) + 2 x \]
    derivative simplify simplify✓ Proved
  7. \[ = 4 x \ln{\left(2 x \right)} \]
    simplify simplify 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 2.895

hard Video
Differentiate \( \displaystyle f(x) = x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \).
  1. \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \]
    derivativeStart with the derivative of the function.✓ Proved
  2. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(4 x \right)} - 1\right) + \left(2 \ln{\left(4 x \right)} - 1\right) \frac{d}{d x} x^{2} \]
    productApply the product rule.✓ Proved
  3. \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(4 x \right)} - 1\right) + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]
    derivativeDifferentiate the first part of the product.✓ Proved
  4. \[ = x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(4 x \right)}\right) + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]
    sumDistribute the derivative over the subtraction.✓ Proved
  5. \[ = 2 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]
    derivativeDifferentiate the terms inside the parentheses.✓ Proved
  6. \[ = 2 x \left(2 \ln{\left(4 x \right)} - 1\right) + \frac{x \frac{d}{d x} 4 x}{2} \]
    chainApply the chain rule to the logarithm.✓ Proved
  7. \[ = 2 x \left(2 \ln{\left(4 x \right)} - 1\right) + 2 x \]
    derivative algebra algebra algebraDifferentiate the inner function 4*x. Simplify the fraction. Simplify the expression. Simplify the second term.✓ Proved
  8. \[ = 4 x \ln{\left(4 x \right)} \]
    algebra algebra algebraFactor out 2*x. Simplify the expression inside the parentheses. Multiply the terms together.✓ Proved
Answer \( 4 x \ln{\left(4 x \right)} \)
Top: the graph of f = x²·(2log(4x) - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x·(2log(4x) - 1) + 2x.
A second solution, written separately

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

  1. \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \]
    derivative✓ Proved
  2. \[ = 2 x \left(2 \ln{\left(4 x \right)} - 1\right) + 2 x \]
    product simplify simplifyApply product rule Simplify the derivative of log(4*x) Simplify x**2*(2/x)✓ Proved
  3. \[ = 4 x \ln{\left(4 x \right)} \]
    simplify simplifyCombine 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

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Problem 2.539medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{3} - x\right) e^{3 x} \).
Problem 2.551medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x \sin{\left(5 x \right)} \).
Problem 2.584medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{5}{3}\right) e^{3 x} \).
Problem 2.622medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{4} - x\right) e^{4 x} \).
Problem 2.632medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{3}\right) e^{3 x + 2} \).
Problem 2.660medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{5}\right) e^{5 x + 2} \).
Problem 2.668medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \left(x - 4\right) e^{x - 3} \).
Problem 2.674medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 - x\right) e^{2 x - 3} \).
Problem 2.676medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{5}{2}\right) e^{2 x} \).
Problem 2.683medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \left(x - 2\right) e^{2 x - 3} \).
Problem 2.710medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x e^{5 x + 1} \).
Problem 2.713medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x - 1\right) e^{5 x} \).
Problem 2.840medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x e^{x + 1} \).
Problem 2.846medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \left(x - 1\right) e^{2 x - 1} \).
Problem 2.866medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(1 - x\right) e^{x} \).
Problem 2.869medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 \left(x - 2\right) e^{2 x - 3} \).
Problem 2.870medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{2}\right) e^{2 x + 2} \).
Problem 2.884medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(- x - 1\right) e^{x + 2} \).
Problem 2.891medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(4 - x\right) e^{x - 3} \).
Problem 2.948medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 \left(x + 1\right) e^{x + 2} \).
Problem 2.951medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(4 x + 1\right) e^{4 x + 2}}{2} \).
Problem 2.960medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{4}{3}\right) e^{3 x - 3} \).
Problem 2.967medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 1\right) e^{2 x - 1} \).
Problem 2.971medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(5 x - 2\right) e^{5 x - 1} \).
Problem 2.982medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \left(x - 2\right) e^{x - 1} \).
Problem 2.1014medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \left(x - 1\right) e^{x} \).
Problem 2.1015medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - \frac{2}{5}\right) e^{5 x} \).
Problem 2.1020medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{2}{3}\right) e^{3 x - 1} \).
Problem 2.1024medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(2 - x\right) e^{x - 1} \).
Problem 2.1066medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{2} - x\right) e^{4 x - 1} \).
Problem 2.1080medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{3}{2}\right) e^{2 x} \).
Problem 2.1117medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \left(x + 1\right) e^{x + 2} \).
Problem 2.1127medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 \left(x - 2\right) e^{2 x - 3} \).
Problem 2.1128medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \left(x - 1\right) e^{2 x - 1} \).
Problem 2.1141medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{3 x + 1} \).
Problem 2.1183medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 4\right) e^{x - 3} \).
Problem 2.1190medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(\frac{1}{5} - x\right) e^{5 x} \).
Problem 2.1198medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 \left(x - 1\right) e^{2 x - 1} \).
Problem 2.1212medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - 1\right) e^{4 x - 1} \).
Problem 2.1219medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{2}{5}\right) e^{5 x - 1} \).
Problem 2.1222medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 \left(x - 1\right) e^{4 x - 3} \).
Problem 2.1223medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{4}\right) e^{4 x + 2} \).
Problem 2.1224medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - \frac{8}{3}\right) e^{3 x - 3} \).
Problem 2.1238medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 2\right) e^{2 x - 3} \).
Problem 2.1244medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 3 \left(x + 1\right) e^{x + 2} \).
Problem 2.1281medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{6}{5}\right) e^{5 x - 1} \).
Problem 2.1286medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + \frac{1}{5}\right) e^{5 x + 2} \).
Problem 2.1295medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 x e^{5 x + 1} \).
Problem 2.1303medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{5}\right) e^{5 x} \).
Problem 2.1316medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(- x - \frac{1}{3}\right) e^{3 x + 2} \).
Problem 2.1320medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x + \frac{3}{5}\right) e^{5 x + 2} \).
Problem 2.1321medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 3 \left(x - 2\right) e^{x - 1} \).
Problem 2.1348medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - 2\right) e^{x - 1} \).
Problem 2.1361medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{3}{5}\right) e^{5 x} \).
Problem 2.1376medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{1}{2}\right) e^{4 x - 1} \).
Problem 2.1380medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{3}{2}\right) e^{4 x - 1} \).
Problem 2.1381medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(5 x - \frac{5}{2}\right) e^{4 x - 1} \).
Problem 2.1386medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - 1\right) e^{3 x} \).
Problem 2.1404medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(3 x - \frac{12}{5}\right) e^{5 x - 3} \).
Problem 2.1420medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(\frac{4}{5} - x\right) e^{5 x - 3} \).
Problem 2.1421medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \left(5 x + \frac{5}{4}\right) e^{4 x + 2} \).
Problem 2.1427medium Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 2 \left(x - 1\right) e^{4 x - 3} \).
Problem 2.1433medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x - \frac{4}{5}\right) e^{5 x - 3} \).
Problem 2.1443medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(2 x - \frac{2}{3}\right) e^{3 x} \).
Problem 2.1445medium Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(x \right)}}{x + 1} \).
Problem 2.316hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
Problem 2.397hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \).
Problem 2.426hard 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 2.478hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{3} e^{x} \).
Problem 2.520hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(3 \ln{\left(2 x \right)} - \frac{3}{2}\right) \).
Problem 2.537hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 9 x^{2} \cos{\left(3 x \right)} \).
Problem 2.575hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
Problem 2.594hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \ln{\left(x \right)} \sin{\left(x \right)} \).
Problem 2.603hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \).
Problem 2.609hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \).
Problem 2.621hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 8 x^{3} e^{2 x} \).
Problem 2.645hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{3} e^{x + 1} \).
Problem 2.652hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \).
Problem 2.653hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \).
Problem 2.656hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x - 1 \right)} + \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
Problem 2.658hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(1 - 2 \ln{\left(4 x \right)}\right) \).
Problem 2.669hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(\ln{\left(x \right)} - \frac{1}{2}\right) \).
Problem 2.839hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{6} \).
Problem 2.844hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
Problem 2.860hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
Problem 2.861hard 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 2.874hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \).
Problem 2.883hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
Problem 2.897hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x^{2} \left(1 - 2 \ln{\left(5 x \right)}\right)}{4} \).
Problem 2.961hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{3} \).
Problem 2.964hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{2} \left(4 \ln{\left(4 x \right)} - 2\right) \).
Problem 2.993hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \).
Problem 2.1018hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{2} \left(5 \ln{\left(5 x \right)} - \frac{5}{2}\right) \).
Problem 2.1022hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(5 \ln{\left(2 x \right)} - \frac{5}{2}\right) \).
Problem 2.1027hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \).
Problem 2.1030hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
Problem 2.1060hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \).
Problem 2.1061hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x - 6\right) \ln{\left(x - 3 \right)} + 3 x + 9 \ln{\left(x - 3 \right)} \).
Problem 2.1064hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{9 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \).
Problem 2.1067hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(3 \ln{\left(3 x \right)} - \frac{3}{2}\right) \).
Problem 2.1086hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{15 x^{2} \left(2 \ln{\left(5 x \right)} - 1\right)}{4} \).
Problem 2.1091hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
Problem 2.1096hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
Problem 2.1109hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{2} \left(\frac{1}{2} - \ln{\left(2 x \right)}\right) \).
Problem 2.1114hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \).
Problem 2.1116hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
Problem 2.1134hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{6} \).
Problem 2.1136hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{4} \).
Problem 2.1178hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.1193hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
Problem 2.1199hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x^{2} + 2 x \left(x + 1\right) \ln{\left(2 x + 1 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2} \).
Problem 2.1208hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{10} \).
Problem 2.1229hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \).
Problem 2.1239hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{2} \).
Problem 2.1242hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
Problem 2.1246hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \).
Problem 2.1249hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{10} \).
Problem 2.1297hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right) e^{2 x - 1}}{4} \).
Problem 2.1307hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{2} \).
Problem 2.1310hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
Problem 2.1312hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{8} \).
Problem 2.1319hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
Problem 2.1327hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 1 \right)} - \cos{\left(2 x + 1 \right)}\right) e^{2 x + 1}}{4} \).
Problem 2.1328hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}}{2} \).
Problem 2.1342hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x^{2} + 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + x + \frac{\ln{\left(2 x - 1 \right)}}{2} \).
Problem 2.1358hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \).
Problem 2.1372hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{3 x - 1} \sin{\left(- 3 x + \frac{\pi}{4} + 1 \right)}}{6} \).
Problem 2.1392hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{3 x - 3} \cos{\left(3 x - 3 + \frac{\pi}{4} \right)}}{6} \).
Problem 2.1402hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x \left(x + 4\right) \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \).
Problem 2.1428hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
Problem 2.1429hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x \right)} - \cos{\left(4 x \right)}\right) e^{4 x}}{8} \).
Problem 2.1435hard 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 2.1447hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{e^{4 x} \cos{\left(4 x \right)}}{8} \).
Problem 2.1489hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x e^{x} - e^{x} \).
Problem 2.1490hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{4 x} - \frac{e^{4 x}}{2} \).
Problem 2.1499hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{2} - 2 x + \left(3 x^{2} + 4 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \).
Problem 2.1512hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} - x + \left(5 x^{2} + 2 x + \frac{1}{5}\right) \ln{\left(5 x + 1 \right)} \).
Problem 2.1515hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x e^{2 x} - \frac{e^{2 x}}{2} \).
Problem 2.1572hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x e^{4 x} - \frac{5 e^{4 x}}{4} \).
Problem 2.1574hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x^{2} + 3 x + \left(2 x^{2} - 6 x + \frac{9}{2}\right) \ln{\left(2 x - 3 \right)} \).
Problem 2.1575hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{4} + \frac{3 x}{2} + \left(\frac{x^{2}}{2} - 3 x + \frac{9}{2}\right) \ln{\left(x - 3 \right)} \).
Problem 2.1577hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x e^{2 x} - \frac{5 e^{2 x}}{2} \).
Problem 2.1582hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + \frac{3 x}{2} + \left(x^{2} - 3 x + \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \).
Problem 2.1587hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{4} - \frac{x}{2} + \left(\frac{x^{2}}{2} + x + \frac{1}{2}\right) \ln{\left(x + 1 \right)} \).
Problem 2.1589hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x e^{3 x} - \frac{e^{3 x}}{3} \).
Problem 2.1590hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x e^{x} - 3 e^{x} \).
Problem 2.1599hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} - 2 x + \left(x^{2} + 4 x + 4\right) \ln{\left(x + 2 \right)} \).
Problem 2.1600hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{4} + \frac{x}{2} + \left(\frac{x^{2}}{2} - x + \frac{1}{2}\right) \ln{\left(x - 1 \right)} \).
Problem 2.1604hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x e^{5 x} - \frac{3 e^{5 x}}{5} \).
Problem 2.1663hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{x} - 2 e^{x} \).
Problem 2.1672hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{3 x} - \frac{2 e^{3 x}}{3} \).
Problem 2.1673hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 5 x e^{5 x} - e^{5 x} \).
Problem 2.1682hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{5 x} - \frac{2 e^{5 x}}{5} \).
Problem 2.1683hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + \frac{x}{2} + \left(x^{2} - x + \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \).
Problem 2.1684hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x e^{5 x} - \frac{e^{5 x}}{5} \).
Problem 2.1693hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} + \ln{\left(3^{\frac{3 x^{2}}{2}} x^{\frac{3 x^{2}}{2}} \right)} \).
Problem 2.1748hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x e^{4 x} + \frac{e^{4 x}}{4} \).
Problem 2.1749hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x e^{4 x} - \frac{e^{4 x}}{4} \).
Problem 2.1751hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} - \frac{x}{2} + \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)} \).
Problem 2.1756hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{3 x} + \frac{e^{3 x}}{3} \).
Problem 2.1781hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x e^{3 x} - e^{3 x} \).
Problem 2.1821hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 3 x e^{2 x} - \frac{3 e^{2 x}}{2} \).
Problem 2.1824hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{5 e^{4 x} \cos{\left(4 x \right)}}{8} \).
Problem 2.1829hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x e^{5 x} + \frac{e^{5 x}}{5} \).
Problem 2.1832hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{5 x^{2} \ln{\left(x \right)}}{2} - \frac{5 x^{2}}{4} \).
Problem 2.1843hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{2 x} + \frac{e^{2 x}}{2} \).
Problem 2.1844hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 e^{4 x} \sin{\left(4 x \right)}}{8} - \frac{3 e^{4 x} \cos{\left(4 x \right)}}{8} \).
Problem 2.1856hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} - 2 x + \left(5 x^{2} + 4 x + \frac{4}{5}\right) \ln{\left(5 x + 2 \right)} \).
Problem 2.1906hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = 2 x e^{2 x} - e^{2 x} \).
Problem 2.1907hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} - 5 x + \left(\frac{5 x^{2}}{2} + 10 x + 10\right) \ln{\left(x + 2 \right)} \).
Problem 2.1909hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{3 x^{2}}{4} + x - \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \).
Problem 2.1913hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{4} - x + \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \).
Problem 2.1922hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} + 3 x + \left(5 x^{2} - 6 x + \frac{9}{5}\right) \ln{\left(5 x - 3 \right)} \).
Problem 2.1930hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{4} - \frac{x}{2} + \left(- \frac{x^{2}}{2} + x - \frac{1}{2}\right) \ln{\left(x - 1 \right)} \).
Problem 2.1931hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x e^{x} + e^{x} \).
Problem 2.1935hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{e^{4 x} \sin{\left(4 x \right)}}{8} + \frac{e^{4 x} \cos{\left(4 x \right)}}{8} \).
Problem 2.1939hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x^{2} - x + \left(2 x^{2} + 2 x + \frac{1}{2}\right) \ln{\left(2 x + 1 \right)} \).
Problem 2.1984hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 5 x^{2} + \frac{5 x}{2} + \left(10 x^{2} - 5 x + \frac{5}{8}\right) \ln{\left(4 x - 1 \right)} \).
Problem 2.1985hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 5 x e^{3 x} - \frac{5 e^{3 x}}{3} \).
Problem 2.1991hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - \frac{3 x}{2} + \left(- x^{2} + 3 x - \frac{9}{4}\right) \ln{\left(2 x - 3 \right)} \).
Problem 2.1996hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{2} - x + \left(3 x^{2} + 2 x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \).
Problem 2.2012hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} + \frac{x}{2} - \left(x^{2} + x + \frac{1}{4}\right) \ln{\left(2 x + 1 \right)} \).
Problem 2.2015hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 5 x^{2} + \frac{15 x}{2} + \left(10 x^{2} - 15 x + \frac{45}{8}\right) \ln{\left(4 x - 3 \right)} \).
Problem 2.2018hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 3 x^{2} + \frac{3 x}{2} + \left(6 x^{2} - 3 x + \frac{3}{8}\right) \ln{\left(4 x - 1 \right)} \).
Problem 2.2019hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - \ln{\left(\left(2 x\right)^{x^{2}} \right)} \).
Problem 2.2049hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 2 x^{2} + x + \left(4 x^{2} - 2 x + \frac{1}{4}\right) \ln{\left(4 x - 1 \right)} \).
Problem 2.2055hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{2} + x + \left(5 x^{2} - 2 x + \frac{1}{5}\right) \ln{\left(5 x - 1 \right)} \).
Problem 2.2060hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + 3 x + \left(x^{2} - 6 x + 9\right) \ln{\left(x - 3 \right)} \).
Problem 2.2072hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} - x + \left(x^{2} + 2 x + 1\right) \ln{\left(x + 1 \right)} \).
Problem 2.2074hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} - \frac{x}{2} + \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \).
Problem 2.2112hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} - \frac{x}{2} + \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)} \).
Problem 2.2113hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} - 3 x + \left(\frac{3 x^{2}}{2} + 6 x + 6\right) \ln{\left(x + 2 \right)} \).
Problem 2.2119hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = x^{2} - \frac{x}{2} + \left(- 2 x^{2} + x - \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \).
Problem 2.2124hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x^{2}}{4} + x - \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \).
Problem 2.2127hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 2 x^{2} - x + \left(4 x^{2} + 2 x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)} \).
Problem 2.2128hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x^{2}}{4} - \frac{x}{2} + \left(- \frac{3 x^{2}}{2} + x - \frac{1}{6}\right) \ln{\left(3 x - 1 \right)} \).
Problem 2.2130hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{e^{4 x} \sin{\left(4 x \right)}}{4} - \frac{e^{4 x} \cos{\left(4 x \right)}}{4} \).
Problem 2.2170hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - x^{2} + \frac{x}{2} + \left(2 x^{2} - x + \frac{1}{8}\right) \ln{\left(4 x - 1 \right)} \).
Problem 2.2262hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 3 x^{2} - \frac{3 x}{2} + \left(6 x^{2} + 3 x + \frac{3}{8}\right) \ln{\left(4 x + 1 \right)} \).
Problem 2.2278hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 5 x^{2} - \frac{5 x}{2} + \left(10 x^{2} + 5 x + \frac{5}{8}\right) \ln{\left(4 x + 1 \right)} \).
Problem 2.2282hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - 2 x^{2} + 3 x + \left(4 x^{2} - 6 x + \frac{9}{4}\right) \ln{\left(4 x - 3 \right)} \).
Problem 2.2283hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} - x + \left(\frac{5 x^{2}}{2} + 2 x + \frac{2}{5}\right) \ln{\left(5 x + 2 \right)} \).
Problem 2.2376hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{4} + x - \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \).
Problem 2.2378hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x^{2}}{4} + \frac{x}{2} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)} \).
Problem 2.2382hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - x^{2} + x + \left(2 x^{2} - 2 x + \frac{1}{2}\right) \ln{\left(2 x - 1 \right)} \).
Problem 2.2383hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = 3 x e^{4 x} - \frac{3 e^{4 x}}{4} \).
Problem 2.2387hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - 3 x^{2} + \frac{9 x}{2} + \left(6 x^{2} - 9 x + \frac{27}{8}\right) \ln{\left(4 x - 3 \right)} \).
Problem 2.2400hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} - x + \left(\frac{3 x^{2}}{2} + 2 x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \).
Problem 2.2414hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{5 x^{2}}{4} - \frac{x}{2} + \left(- \frac{5 x^{2}}{2} + x - \frac{1}{10}\right) \ln{\left(5 x - 1 \right)} \).
Problem 2.2422hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{2} - \frac{3 x}{2} + \left(- 2 x^{2} + 3 x - \frac{9}{8}\right) \ln{\left(4 x - 3 \right)} \).
Problem 2.2426hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = x^{2} + \frac{x}{2} - \left(2 x^{2} + x + \frac{1}{8}\right) \ln{\left(4 x + 1 \right)} \).
Problem 2.2427hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + x + \left(x^{2} - 2 x + 1\right) \ln{\left(x - 1 \right)} \).
Problem 2.2462hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{2} + x + \left(3 x^{2} - 2 x + \frac{1}{3}\right) \ln{\left(3 x - 1 \right)} \).
Problem 2.2477hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = \frac{3 x^{2}}{4} + \frac{x}{2} - \left(\frac{3 x^{2}}{2} + x + \frac{1}{6}\right) \ln{\left(3 x + 1 \right)} \).
Problem 2.2534hard Video✓ Every equation proved
Differentiate \( \displaystyle f(x) = - \frac{5 x^{2}}{4} + \frac{x}{2} + \left(\frac{5 x^{2}}{2} - x + \frac{1}{10}\right) \ln{\left(5 x - 1 \right)} \).
Problem 2.2555hard Video✓ Nihil obstat
Differentiate \( \displaystyle f(x) = - \frac{25 x^{2}}{4} - 5 x + \left(\frac{25 x^{2}}{2} + 10 x + 2\right) \ln{\left(5 x + 2 \right)} \).
Problem 2.2556hard Video✓ Every equation proved