The derivative of a quotient: (f/g)′ = (f′g − fg′)/g². 119 problems with worked solutions; in 119 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
Differentiate \( \displaystyle f(x) = \frac{\sqrt{4 x + 1}}{4 x + 2} \).
\[ \frac{d}{d x} \frac{\sqrt{4 x + 1}}{4 x + 2} \]
rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
\[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sqrt{4 x + 1}}{4 x + 2} \]
productrewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
\[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainApply the chain rule to the first term.✓ Proved
\[ = - \frac{4 \sqrt{4 x + 1}}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainconstant-multiplealgebraApply the chain rule to the second term. Simplify the constants. Distribute the exponents.✓ Proved
\[ = - \frac{8 x}{\sqrt{4 x + 1} \left(4 x + 2\right)^{2}} \]
algebraalgebraalgebrasimplifyFind a common denominator. Simplify the numerator. Distribute the 4. Combine like terms.✓ Proved
Answer \( - \frac{2 x}{\left(2 x + 1\right)^{2} \sqrt{4 x + 1}} \)
Top: the graph of f = sqrt(4x + 1)/(4x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4sqrt(4x + 1)/(4x + 2)² + 2/(sqrt(4x + 1)·(4x + 2)).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sqrt{4 x + 1}}{4 x + 2} \]
derivativerewriterewriteStart by identifying the function to differentiate. Rewrite the square root as a power of 1/2 for easier handling. Rewrite the division as multiplication by a negative power.✓ Proved
\[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sqrt{4 x + 1}}{4 x + 2} \]
productApply the product rule to the two factors.✓ Proved
\[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \left(4 x + 1\right)}{2 \sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainApply the chain rule to the first factor (4*x + 1)**(1/2).✓ Proved
\[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainApply the chain rule to the second factor (4*x + 2)**(-1).✓ Proved
\[ = - \frac{\sqrt{4 x + 1} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainApply the chain rule to differentiate the inner function 4*x + 2.✓ Proved
\[ = - \frac{4 \sqrt{4 x + 1}}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
chainsimplifyrewriterewriteDifferentiate the inner linear function 4*x + 2 to get 4. Simplify the constants: (1/2)*4 = 2 and -1*4 = -4. Rewrite negative exponents as fractions. Rewrite the fractional exponent as a square root.✓ Proved
\[ = - \frac{8 x}{\sqrt{4 x + 1} \left(4 x + 2\right)^{2}} \]
algebraalgebrasimplifysimplifyFind a common denominator to combine the two terms. Expand the numerator terms. Combine like terms in the numerator: 8x - 16x = -8x and 4 - 4 = 0. Final simplified form of the 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
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x + 1 \right)}}{2 x + 1} \).
\[ \frac{d}{d x} \frac{\sin{\left(2 x + 1 \right)}}{2 x + 1} \]
rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
\[ = \sin{\left(2 x + 1 \right)} \frac{d}{d x} \frac{1}{2 x + 1} + \frac{\frac{d}{d x} \sin{\left(2 x + 1 \right)}}{2 x + 1} \]
productApply the product rule.✓ Proved
\[ = \sin{\left(2 x + 1 \right)} \frac{d}{d x} \frac{1}{2 x + 1} + \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} \]
chainconstant-multipleDifferentiate the first term using the chain rule. Factor out the constant 2.✓ Proved
\[ = \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} - \frac{\sin{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{\left(2 x + 1\right)^{2}} \]
chainDifferentiate the second term using the chain rule.✓ Proved
\[ = \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} - \frac{2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \]
derivativealgebraalgebraDifferentiate the inner function 2*x + 1. Simplify the expression by multiplying the constants. Rewrite the negative exponents as denominators.✓ Proved
\[ = \frac{2 \left(2 x + 1\right) \cos{\left(2 x + 1 \right)} - 2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \]
algebraalgebraCombine the terms over a common denominator. Factor out the common factor 2.✓ Proved
Answer \( \frac{2 \cos{\left(2 x + 1 \right)}}{2 x + 1} - \frac{2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \)
Top: the graph of f = sin(2x + 1)/(2x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2cos(2x + 1)/(2x + 1) - 2sin(2x + 1)/(2x + 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sin{\left(2 x + 1 \right)}}{2 x + 1} \]
derivative✓ Proved
\[ = \frac{2 \left(2 x + 1\right) \cos{\left(2 x + 1 \right)} - 2 \sin{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \]
quotientalgebra✓ 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
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \).
\[ \frac{d}{d x} \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \sin{\left(4 x + 1 \right)} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sin{\left(4 x + 1 \right)}}{4 x + 2} \]
productApply the product rule.✓ Proved
\[ = \sin{\left(4 x + 1 \right)} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} \]
chainApply the chain rule to the first term.✓ Proved
\[ = \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} - \frac{\sin{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2}} \]
powerApply the power rule to the second term.✓ Proved
\[ = \frac{4 \cos{\left(4 x + 1 \right)}}{4 x + 2} - \frac{4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
chainalgebraalgebraApply the chain rule to the inner function. Simplify the expression by combining constants. Rewrite with negative exponents as fractions.✓ Proved
\[ = \frac{4 \left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
algebraFind a common denominator.✓ Proved
\[ = \frac{16 x \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)} + 8 \cos{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
algebraDistribute the term in the numerator.✓ Proved
\[ = \frac{\left(16 x + 8\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
simplifyFactor out the common constant 4.✓ Proved
Answer \( \frac{\left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - \sin{\left(4 x + 1 \right)}}{\left(2 x + 1\right)^{2}} \)
Top: the graph of f = sin(4x + 1)/(4x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4cos(4x + 1)/(4x + 2) - 4sin(4x + 1)/(4x + 2)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sin{\left(4 x + 1 \right)}}{4 x + 2} \]
derivativeStart with the derivative of the quotient.✓ Proved
\[ = \frac{4 \left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
quotientconstant-multipleconstant-multipleApply the quotient rule. Factor out the constant 4. Take the 4 outside the parentheses.✓ Proved
\[ = \frac{16 x \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)} + 8 \cos{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
algebraDistribute cos over the product (4*x+2).✓ Proved
\[ = \frac{4 \left(4 x + 2\right) \cos{\left(4 x + 1 \right)} - 4 \sin{\left(4 x + 1 \right)}}{\left(4 x + 2\right)^{2}} \]
algebraCombine the terms back into a single product.✓ Proved
\[ = \frac{4 \cos{\left(4 x + 1 \right)} - \frac{4 \sin{\left(4 x + 1 \right)}}{4 x + 2}}{4 x + 2} \]
simplifyCancel one factor of (4*x+2) from numerator and denominator.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebrasimplifyDistribute and simplify the numerator terms. Combine like terms in the numerator.✓ 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
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(2 x - 1 \right)}}{2 x - 1} \).
\[ \frac{d}{d x} \frac{\sin{\left(2 x - 1 \right)}}{2 x - 1} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}{2 x - 1} \]
productApply the product rule.✓ Proved
\[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{\cos{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]
chainApply the chain rule to the first term.✓ Proved
\[ = \sin{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} \]
derivativeDifferentiate the inner function 2*x - 1.✓ Proved
\[ = \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{\sin{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\left(2 x - 1\right)^{2}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
derivativealgebraDifferentiate the inner function 2*x - 1. Simplify the terms using exponent rules.✓ Proved
\[ = \frac{2 \left(2 x - 1\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
algebraCombine the terms over a common denominator.✓ Proved
\[ = \frac{\left(4 x - 2\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
simplifyFinal simplified form.✓ Proved
Answer \( \frac{2 \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \)
Top: the graph of f = sin(2x - 1)/(2x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2cos(2x - 1)/(2x - 1) - 2sin(2x - 1)/(2x - 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sin{\left(2 x - 1 \right)}}{2 x - 1} \]
derivativeStart with the derivative of the quotient✓ Proved
\[ = \frac{2 \left(2 x - 1\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
quotientconstant-multipleApply the quotient rule Factor out the common factor 2✓ Proved
\[ = \frac{\left(4 x - 2\right) \cos{\left(2 x - 1 \right)} - 2 \sin{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
simplifyDistribute the 2 inside the numerator for a simpler form✓ 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
Top: the graph of f = (x² - 1)/(x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2x·(x² - 1)/(x² + 1)² + 2x/(x² + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{x^{2} - 1}{x^{2} + 1} \]
derivative✓ Proved
\[ = \frac{- 2 x \left(x^{2} - 1\right) + 2 x \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}} \]
quotientconstant-multipleApply the quotient rule Factor out 2*x✓ Proved
\[ = \frac{4 x}{\left(x^{2} + 1\right)^{2}} \]
algebrasimplifysimplifySubtract the parentheses Simplify inside the parentheses Combine constants✓ 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
chainalgebraalgebrasimplifyDifferentiate the power term using the chain rule. Simplify the terms using exponent rules. Simplify the coefficients. Simplify the fraction.✓ Proved
\[ = \frac{2 x e^{2 x} - e^{2 x}}{2 x^{2}} \]
algebraCombine the terms over a common denominator.✓ Proved
\[ = \frac{\left(2 x - 1\right) e^{2 x}}{2 x^{2}} \]
Top: the graph of f = e^(2x)/(2x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(2x)/x - e^(2x)/(2x²).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{e^{2 x}}{2 x} \]
derivativerewriteStart by identifying the function to differentiate. Rewrite the denominator as a negative power to prepare for the product rule.✓ Proved
powersimplifysimplifyDifferentiate the second term using the power rule and chain rule. Simplify the coefficients in each term. Simplify the fraction 2/4 to 1/2.✓ Proved
\[ = \frac{2 x e^{2 x} - e^{2 x}}{2 x^{2}} \]
algebraCombine the two terms over a common denominator.✓ Proved
\[ = \frac{\left(2 x - 1\right) e^{2 x}}{2 x^{2}} \]
simplifyFactor out exp(2*x) from the numerator.✓ 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
Top: the graph of f = e^(x)/(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(x)/(x + 1) - e^(x)/(x + 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \).
\[ \frac{d}{d x} \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \]
derivativequotientStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \cos{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{2 x + 2} + \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} \]
productApply the product rule.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{\cos{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{\left(2 x + 2\right)^{2}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
derivativeDifferentiate the inner function 2*x + 2.✓ Proved
\[ = - \frac{2 \sin{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
chainalgebraalgebraApply the chain rule to the first term. Distribute and simplify the terms. Rewrite negative exponents as denominators.✓ Proved
\[ = \frac{- 2 \left(2 x + 2\right) \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
algebrasimplifyFind a common denominator. Factor out the common term -2.✓ Proved
Answer \( \frac{\left(- 2 x - 2\right) \sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}}{2 \left(x + 1\right)^{2}} \)
Top: the graph of f = cos(2x + 2)/(2x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2sin(2x + 2)/(2x + 2) - 2cos(2x + 2)/(2x + 2)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \]
derivative✓ Proved
\[ = \frac{- 2 \left(2 x + 2\right) \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
quotientalgebraalgebra✓ 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
constant-multiplerewritealgebraPull out the constant factor 5/2. Rewrite the inner log for clarity. Simplify the expression.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(2 x + 2 \right)}}{2 \ln{\left(2 x + 2 \right)}} \]
chainApply the chain rule to the outer logarithm.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(2 x + 2\right)}{2 \left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \]
chainApply the chain rule to the inner logarithm.✓ Proved
\[ = \frac{5}{\left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \]
derivativealgebraalgebrasimplifyDifferentiate the innermost linear function. Simplify the product. Simplify the fraction. Final simplified form.✓ Proved
Answer \( \frac{5}{\left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \)
Top: the graph of f = 5log(log(2x + 2))/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/((2x + 2)·log(2x + 2)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
constant-multiplePull out the constant 5/2✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(2 x + 2 \right)}}{2 \ln{\left(2 x + 2 \right)}} \]
chainDerivative of log(log(...))✓ Proved
\[ = \frac{5}{\left(2 x + 2\right) \ln{\left(2 x + 2 \right)}} \]
chainsimplifyDerivative of log(2*x + 2) Simplify 2/(2*x+2) to 1/(x+1) and combine constants✓ 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
Differentiate \( \displaystyle f(x) = \frac{\sin{\left(4 x \right)}}{4 x} \).
\[ \frac{d}{d x} \frac{\sin{\left(4 x \right)}}{4 x} \]
rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
\[ = \sin{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} + \frac{\frac{d}{d x} \sin{\left(4 x \right)}}{4 x} \]
productApply the product rule.✓ Proved
\[ = \sin{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} + \frac{\cos{\left(4 x \right)}}{x} \]
chainDifferentiate the first term using the chain rule.✓ Proved
\[ = \frac{\cos{\left(4 x \right)}}{x} - \frac{\sin{\left(4 x \right)}}{4 x^{2}} \]
chainalgebraalgebraalgebrasimplifyDifferentiate the second term using the chain rule. Simplify the signs and constants. Rewrite the negative exponents as denominators. Simplify the fractions. Simplify the final expression.✓ Proved
Answer \( \frac{\cos{\left(4 x \right)}}{x} - \frac{\sin{\left(4 x \right)}}{4 x^{2}} \)
Top: the graph of f = sin(4x)/(4x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = cos(4x)/x - sin(4x)/(4x²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\sin{\left(4 x \right)}}{4 x} \]
derivative✓ Proved
\[ = \frac{16 x \cos{\left(4 x \right)} - 4 \sin{\left(4 x \right)}}{16 x^{2}} \]
quotientsimplify✓ Proved
\[ = \frac{4 x \cos{\left(4 x \right)} - \sin{\left(4 x \right)}}{4 x^{2}} \]
algebra✓ 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
Top: the graph of f = (4x² - 1)/(4x² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -8x·(4x² - 1)/(4x² + 1)² + 8x/(4x² + 1).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
quotientApply the quotient rule: (u'v - uv') / v^2.✓ Proved
\[ = \frac{- 8 x \left(4 x^{2} - 1\right) + 8 x \left(4 x^{2} + 1\right)}{\left(4 x^{2} + 1\right)^{2}} \]
powerDifferentiate the inner polynomial terms using the power rule.✓ Proved
\[ = \frac{16 x}{\left(4 x^{2} + 1\right)^{2}} \]
algebraalgebrasimplifyExpand the numerator terms. Distribute the negative sign in the second term. Combine like terms in the numerator.✓ 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
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(4 x \right)}}{4 x} \).
\[ \frac{d}{d x} \frac{\cos{\left(4 x \right)}}{4 x} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \cos{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} + \frac{\frac{d}{d x} \cos{\left(4 x \right)}}{4 x} \]
productApply the product rule.✓ Proved
\[ = \cos{\left(4 x \right)} \frac{d}{d x} \frac{1}{4 x} - \frac{\sin{\left(4 x \right)}}{x} \]
chainconstant-multipleApply the chain rule to the cosine term. Factor out the constant 4.✓ Proved
\[ = - \frac{\sin{\left(4 x \right)}}{x} - \frac{\cos{\left(4 x \right)}}{4 x^{2}} \]
chainalgebraalgebrasimplifyApply the chain rule to the power term. Simplify the signs and distribute the terms. Rewrite the negative exponents as fractions. Cancel the common factor of 4 in the first term.✓ Proved
Answer \( - \frac{\sin{\left(4 x \right)}}{x} - \frac{\cos{\left(4 x \right)}}{4 x^{2}} \)
Top: the graph of f = cos(4x)/(4x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -sin(4x)/x - cos(4x)/(4x²).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\cos{\left(4 x \right)}}{4 x} \]
derivative✓ Proved
\[ = \frac{- 16 x \sin{\left(4 x \right)} - 4 \cos{\left(4 x \right)}}{16 x^{2}} \]
quotientsimplifyApply the quotient rule to cos(4*x)/(4*x). Multiply out the numerator.✓ Proved
\[ = \frac{- 4 x \sin{\left(4 x \right)} - \cos{\left(4 x \right)}}{4 x^{2}} \]
simplifyFactor out 4 from numerator and denominator.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
derivativealgebraDifferentiate the second term. Simplify the terms.✓ Proved
\[ = \frac{2 - x}{2 x^{2} \sqrt{x - 1}} \]
algebrasimplifyFind a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{2 - x}{2 x^{2} \sqrt{x - 1}} \)
Top: the graph of f = sqrt(x - 1)/x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 1/(2x·sqrt(x - 1)) - sqrt(x - 1)/x².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
constantsimplifysimplifyDerivative of (x-1) is 1. Simplify the fractions. Rewrite powers as sqrt.✓ 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
constant-multiplerewritesimplifyPull out the constant factor. Rewrite the inner log for clarity. Simplify the expression.✓ Proved
\[ = \frac{5 \frac{d}{d x} \ln{\left(3 x - 3 \right)}}{3 \ln{\left(3 x - 3 \right)}} \]
chainApply the chain rule to the outer logarithm.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(3 x - 3\right)}{3 \left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \]
chainApply the chain rule to the inner logarithm.✓ Proved
\[ = \frac{5}{\left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \]
derivativealgebrasimplifyDifferentiate the innermost linear function. Multiply the terms together. Cancel the 3 in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \)
Top: the graph of f = 5log(log(3x - 3))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/((3x - 3)·log(3x - 3)).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ = \frac{5 \frac{d}{d x} \ln{\left(3 x - 3 \right)}}{3 \ln{\left(3 x - 3 \right)}} \]
chain✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(3 x - 3\right)}{3 \left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \]
chain✓ Proved
\[ = \frac{5}{\left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \]
derivativesimplify✓ 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
Top: the graph of f = e^(x)/x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = e^(x)/x - e^(x)/x².
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{e^{x}}{x} \]
derivativealgebraStart by identifying the function to differentiate. Rewrite the division as multiplication by a negative power to prepare for the product rule.✓ Proved
productApply the product rule: (uv)' = u'v + uv'.✓ Proved
\[ = \frac{e^{x}}{x} - \frac{e^{x}}{x^{2}} \]
chainalgebraDifferentiate exp(x) to get exp(x) and x^(-1) to get -x^(-2) using the power rule. Rewrite negative exponents as fractions for clarity.✓ Proved
\[ = \frac{x e^{x} - e^{x}}{x^{2}} \]
simplifyCombine the terms over a common denominator.✓ Proved
\[ = \frac{\left(x - 1\right) e^{x}}{x^{2}} \]
algebraFactor out exp(x) from the numerator.✓ 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
chainApply the chain rule to the exponential term.✓ Proved
\[ = - x e^{- x} + e^{- x} \]
derivativealgebraDifferentiate the exponent. Simplify the expression.✓ Proved
\[ = \left(1 - x\right) e^{- x} \]
simplifyFactor out the common term.✓ Proved
Answer \( \left(1 - x\right) e^{- x} \)
Top: the graph of f = x·e^(-x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -x·e^(-x) + e^(-x).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} x e^{- x} \]
derivative✓ Proved
\[ = x \frac{d}{d x} e^{- x} + e^{- x} \frac{d}{d x} x \]
product✓ Proved
\[ = - x e^{- x} + e^{- x} \]
derivativesimplify✓ Proved
\[ = \left(1 - x\right) e^{- x} \]
algebra✓ 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
derivativeDifferentiate the remaining terms.✓ Proved
\[ = \left(3 - x\right) e^{3 - x} + e^{3 - x} \]
algebraDistribute the negative sign.✓ Proved
\[ = \left(4 - x\right) e^{3 - x} \]
algebraalgebrasimplifyFactor out the common exponential term. Simplify the expression inside the parentheses. Final simplified form.✓ Proved
Answer \( \left(4 - x\right) e^{3 - x} \)
Top: the graph of f = (x - 3)·e^(3 - x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(x - 3)·e^(3 - x) + e^(3 - x).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x - 3\right) e^{3 - x} \]
derivativeStart by identifying the function to differentiate.Not checked
derivativealgebraDifferentiate the inner function 3 - x to get -1. Simplify the multiplication by -1.✓ Proved
\[ = \left(4 - x\right) e^{3 - x} \]
algebrasimplifyFactor out the common term exp(3 - x). Simplify the expression inside the parentheses: 1 - x + 3 = 4 - 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
chainderivativealgebraDifferentiate the second part using the chain rule. Differentiate the power term. Simplify the expression by distributing the terms.✓ Proved
Top: the graph of f = cos(x + 2)/(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -sin(x + 2)/(x + 2) - cos(x + 2)/(x + 2)².
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
algebraRearrange the terms in the numerator for clarity.✓ 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
Differentiate \( \displaystyle f(x) = \frac{\cos{\left(2 x - 1 \right)}}{2 x - 1} \).
\[ \frac{d}{d x} \frac{\cos{\left(2 x - 1 \right)}}{2 x - 1} \]
derivativerewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \cos{\left(2 x - 1 \right)} \frac{d}{d x} \frac{1}{2 x - 1} + \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{2 x - 1} \]
productApply the product rule.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{\cos{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\left(2 x - 1\right)^{2}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x - 1 \right)}}{2 x - 1} - \frac{2 \cos{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
derivativeDifferentiate the inner function 2*x - 1.✓ Proved
\[ = - \frac{2 \sin{\left(2 x - 1 \right)}}{2 x - 1} - \frac{2 \cos{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
trigalgebraDifferentiate the cosine function. Simplify the expression by distributing terms.✓ Proved
\[ = \frac{- 2 \left(2 x - 1\right) \sin{\left(2 x - 1 \right)} - 2 \cos{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
simplifyCombine the terms over a common denominator.✓ Proved
Answer \( \frac{2 \left(\left(1 - 2 x\right) \sin{\left(2 x - 1 \right)} - \cos{\left(2 x - 1 \right)}\right)}{\left(2 x - 1\right)^{2}} \)
Top: the graph of f = cos(2x - 1)/(2x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2sin(2x - 1)/(2x - 1) - 2cos(2x - 1)/(2x - 1)².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \frac{\cos{\left(2 x - 1 \right)}}{2 x - 1} \]
derivative✓ Proved
\[ = \frac{- 2 \left(2 x - 1\right) \sin{\left(2 x - 1 \right)} - 2 \cos{\left(2 x - 1 \right)}}{\left(2 x - 1\right)^{2}} \]
quotientsimplifyalgebrasimplifyFactor out the common -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