ln x and ln of anything, where the chain rule gives u′/u. 219 problems with worked solutions; in 219 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 \( \displaystyle f(x) = x \ln{\left(2 x - 3 \right)} - x - \frac{3 \ln{\left(2 x - 3 \right)}}{2} \).
\[ \frac{d}{d x} \left(x \ln{\left(2 x - 3 \right)} - x - \frac{3 \ln{\left(2 x - 3 \right)}}{2}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(2 x - 3 \right)}}{2} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(2 x - 3 \right)}}{2} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} - 1 \]
constant-multiplePull out the constant factor 3/2.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - 1 - \frac{3 \frac{d}{d x} \left(2 x - 3\right)}{2 \left(2 x - 3\right)} \]
chainApply the chain rule to the logarithm term.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
derivativealgebraThe derivative of 2*x - 3 is 2. Simplify the product of the constants.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} \frac{d}{d x} x - 1 - \frac{3}{2 x - 3} \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
derivativeThe derivative of x is 1.✓ Proved
\[ = \frac{x \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
chainApply the chain rule to the logarithm term again.✓ Proved
\[ = \frac{2 x}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
derivativealgebraThe derivative of 2*x - 3 is 2. Simplify the term with x.✓ Proved
\[ = \ln{\left(2 x - 3 \right)} \]
algebraalgebrasimplifyCombine the fractions with the same denominator. Simplify the fraction (2x-3)/(2x-3). Final simplification.✓ Proved
Answer \( \log{\left(2 x - 3 \right)} \)
Top: the graph of f = x·log(2x - 3) - x - 3log(2x - 3)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x/(2x - 3) + log(2x - 3) - 1 - 3/(2x - 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(2 x - 3 \right)} - x - \frac{3 \ln{\left(2 x - 3 \right)}}{2}\right) \]
derivative✓ Proved
\[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} x \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- \frac{3 \ln{\left(2 x - 3 \right)}}{2}\right) \]
sum✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} + \frac{d}{d x} \left(- \frac{3 \ln{\left(2 x - 3 \right)}}{2}\right) - 1 \]
derivative✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(2 x - 3 \right)}}{2} - 1 \]
constant-multiple✓ Proved
\[ = \frac{d}{d x} x \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
chain✓ Proved
\[ = x \frac{d}{d x} \ln{\left(2 x - 3 \right)} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
product✓ Proved
\[ = \frac{2 x}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 - \frac{3}{2 x - 3} \]
chainsimplify✓ Proved
\[ = \ln{\left(2 x - 3 \right)} \]
simplifysimplifysimplify✓ 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
algebraalgebraCombine the fractions. Factor the numerator of the fraction.✓ Proved
\[ = 3 \ln{\left(x - 1 \right)} \]
simplifysimplifySimplify the fraction (x-1)/(x-1) to 1. Combine the constant terms.✓ Proved
Answer \( 3 \log{\left(x - 1 \right)} \)
Top: the graph of f = 3x·log(x - 1) - 3x - 3log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x/(x - 1) + 3log(x - 1) - 3 - 3/(x - 1).
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(3 x \ln{\left(x - 1 \right)} - 3 x - 3 \ln{\left(x - 1 \right)}\right) \]
derivativeStart by expressing the problem as the derivative of the given function with respect to x.Not checked
productalgebraApply the product rule to 3*x*log(x-1) and the chain/derivative rule for log(x-1). Simplify the fractions and remove parentheses.✓ Proved
algebraCombine the terms with denominator (x-1).✓ Proved
\[ = 3 \ln{\left(x - 1 \right)} \]
algebrasimplifySimplify the fraction (3x-3)/(x-1) to 3. Cancel the +3 and -3 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
algebraalgebraCombine the fractions. Factor the numerator.✓ Proved
\[ = 5 \ln{\left(x + 2 \right)} \]
algebrasimplifySimplify the fraction. Combine the remaining terms.✓ Proved
Answer \( 5 \ln{\left(x + 2 \right)} \)
Top: the graph of f = -5x + (5x + 10)·log(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5log(x + 2) - 5 + (5x + 10)/(x + 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- 5 x + \left(5 x + 10\right) \ln{\left(x + 2 \right)}\right) \]
✓ 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
algebraalgebraCombine the fractions. Factor the numerator.✓ Proved
\[ = 5 \ln{\left(x - 3 \right)} \]
algebrasimplifySimplify the fraction. Combine the constant terms.✓ Proved
Answer \( 5 \ln{\left(x - 3 \right)} \)
Top: the graph of f = -5x + (5x - 15)·log(x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5log(x - 3) - 5 + (5x - 15)/(x - 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- 5 x + \left(5 x - 15\right) \ln{\left(x - 3 \right)}\right) \]
derivativesimplifyDifferentiate 5*x-15 and log(x-3). Rewrite the product of 1/(x-3).✓ Proved
\[ = 5 \ln{\left(x - 3 \right)} \]
simplifysimplifySimplify (5*x-15)/(x-3) to 5. Cancel -5 and +5.✓ 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
algebraSimplify the fraction (2*x)/(2*x + 2) by dividing numerator and denominator by 2.✓ Proved
\[ = \ln{\left(2 x + 2 \right)} \]
algebraalgebrasimplifyCombine the fractions with the same denominator. Simplify the fraction (x + 1)/(x + 1) to 1. Combine the remaining terms.✓ Proved
Answer \( \log{\left(2 x + 2 \right)} \)
Top: the graph of f = x·log(2x + 2) - x + log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x/(2x + 2) + log(2x + 2) - 1 + 1/(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(2 x + 2 \right)} - x + \ln{\left(x + 1 \right)}\right) \]
derivative✓ Proved
\[ = x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + \ln{\left(2 x + 2 \right)} - 1 + \frac{1}{x + 1} \]
productApply product rule to x*log(2*x+2) and differentiate -x and log(x+1).✓ Proved
\[ = \frac{x \frac{d}{d x} \left(2 x + 2\right)}{2 x + 2} + \ln{\left(2 x + 2 \right)} - 1 + \frac{1}{x + 1} \]
chainDifferentiate log(2*x+2) using chain rule.✓ Proved
\[ = \frac{2 x}{2 x + 2} + \ln{\left(2 x + 2 \right)} - 1 + \frac{1}{x + 1} \]
constant-multipleSimplify derivative of 2*x+2.✓ Proved
simplifysimplifysimplifyCombine fractions. Simplify (x+1)/(x+1) to 1. Cancel +1 and -1.✓ 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 \( \displaystyle f(x) = x \ln{\left(3 x + 1 \right)} - x + \frac{\ln{\left(3 x + 1 \right)}}{3} \).
\[ \frac{d}{d x} \left(x \ln{\left(3 x + 1 \right)} - x + \frac{\ln{\left(3 x + 1 \right)}}{3}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(3 x + 1 \right)}}{3} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(3 x + 1 \right)}}{3} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(3 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 1 \]
constant-multiplePull out the constant factor 1/3.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{x \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} + \ln{\left(3 x + 1 \right)} - 1 + \frac{\frac{d}{d x} \left(3 x + 1\right)}{3 \left(3 x + 1\right)} \]
chainApply the chain rule to the log terms.✓ Proved
\[ = \frac{3 x}{3 x + 1} + \ln{\left(3 x + 1 \right)} - 1 + \frac{1}{3 x + 1} \]
derivativealgebraThe derivative of 3*x + 1 is 3. Simplify the fractions.✓ Proved
\[ = \ln{\left(3 x + 1 \right)} \]
algebraalgebrasimplifyCombine the fractions. Simplify the fraction (3x+1)/(3x+1) to 1. Final simplification.✓ Proved
Answer \( \ln{\left(3 x + 1 \right)} \)
Top: the graph of f = x·log(3x + 1) - x + log(3x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x/(3x + 1) + log(3x + 1) - 1 + 1/(3x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(3 x + 1 \right)} - x + \frac{\ln{\left(3 x + 1 \right)}}{3}\right) \]
derivative✓ Proved
\[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(3 x + 1 \right)}}{3} \]
sum✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \frac{\ln{\left(3 x + 1 \right)}}{3} \]
product✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(3 x + 1 \right)}}{3} - 1 \]
derivative✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 1 \]
constant-multiple✓ Proved
\[ = \frac{x \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} + \ln{\left(3 x + 1 \right)} - 1 + \frac{\frac{d}{d x} \left(3 x + 1\right)}{3 \left(3 x + 1\right)} \]
logarithmic✓ Proved
\[ = \frac{3 x}{3 x + 1} + \ln{\left(3 x + 1 \right)} - 1 + \frac{1}{3 x + 1} \]
derivativesimplifyalgebra✓ Proved
\[ = \ln{\left(3 x + 1 \right)} \]
simplifysimplifysimplify✓ 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
algebraalgebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
\[ = 5 \ln{\left(x + 1 \right)} \]
simplifysimplifySimplify the fraction. Final simplification.✓ Proved
Answer \( 5 \ln{\left(x + 1 \right)} \)
Top: the graph of f = 5x·log(x + 1) - 5x + 5log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(x + 1) + 5log(x + 1) - 5 + 5/(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(5 x \ln{\left(x + 1 \right)} - 5 x + 5 \ln{\left(x + 1 \right)}\right) \]
simplifysimplifyCancel 5 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
Differentiate \( \displaystyle f(x) = x \ln{\left(3 x - 1 \right)} - x - \frac{\ln{\left(3 x - 1 \right)}}{3} \).
\[ \frac{d}{d x} \left(x \ln{\left(3 x - 1 \right)} - x - \frac{\ln{\left(3 x - 1 \right)}}{3}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} \]
sumApply the sum rule to separate the terms.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(3 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(3 x - 1 \right)}}{3} - 1 \]
constant-multiplePull out the constant factor 1/3.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} x - \frac{\frac{d}{d x} \ln{\left(3 x - 1 \right)}}{3} - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(3 x - 1 \right)}}{3} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{x \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} + \ln{\left(3 x - 1 \right)} - 1 - \frac{\frac{d}{d x} \left(3 x - 1\right)}{3 \left(3 x - 1\right)} \]
chainApply the chain rule to the log terms.✓ Proved
\[ = \frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)} - 1 - \frac{1}{3 x - 1} \]
derivativealgebraThe derivative of 3x - 1 is 3. Simplify the fractions.✓ Proved
\[ = \ln{\left(3 x - 1 \right)} \]
algebrasimplifysimplifyCombine the terms with the same denominator. Simplify the expression. Final result.✓ Proved
Answer \( \ln{\left(3 x - 1 \right)} \)
Top: the graph of f = x·log(3x - 1) - x - log(3x - 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3x/(3x - 1) + log(3x - 1) - 1 - 1/(3x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(3 x - 1 \right)} - x - \frac{\ln{\left(3 x - 1 \right)}}{3}\right) \]
derivative✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} \]
sum✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} x - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} - 1 \]
product✓ Proved
\[ = x \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} - 1 \]
constantsimplify✓ Proved
\[ = \frac{x \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} + \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} - 1 \]
chain✓ Proved
\[ = \frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(3 x - 1 \right)}}{3} - 1 \]
derivativesimplify✓ Proved
\[ = \frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(3 x - 1 \right)}}{3} - 1 \]
constant-multiple✓ Proved
\[ = \frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)} - 1 - \frac{\frac{d}{d x} \left(3 x - 1\right)}{3 \left(3 x - 1\right)} \]
chain✓ Proved
\[ = \frac{3 x}{3 x - 1} + \ln{\left(3 x - 1 \right)} - 1 - \frac{1}{3 x - 1} \]
derivativesimplify✓ Proved
\[ = \ln{\left(3 x - 1 \right)} \]
simplifysimplifyCombine the rational terms Remove the zero term✓ 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
algebraalgebraMultiply the terms in the second part. Simplify the fraction.✓ Proved
\[ = \ln{\left(2 x - 3 \right)} \]
derivativesimplifyDifferentiate the first term. Combine the constants.✓ Proved
Answer \( \ln{\left(2 x - 3 \right)} \)
Top: the graph of f = -x + (x - 3/2)·log(2x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2(x - 3/2)/(2x - 3) + log(2x - 3) - 1.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- x + \left(x - \frac{3}{2}\right) \ln{\left(2 x - 3 \right)}\right) \]
derivativesimplifyDerivative of x-3/2 is 1✓ Proved
\[ = \frac{2 \left(x - \frac{3}{2}\right)}{2 x - 3} + \ln{\left(2 x - 3 \right)} - 1 \]
chainDerivative of log(2*x-3) is 2/(2*x-3)✓ Proved
\[ = \ln{\left(2 x - 3 \right)} \]
simplifysimplifysimplifysimplifysimplifyCombine -1 and +1✓ 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
algebrasimplifyMultiply 4 by (x + 1/4) to get (4x + 1). Simplify the fraction.✓ Proved
\[ = \ln{\left(4 x + 1 \right)} \]
derivativesimplifyDifferentiate the first term. Combine the constants.✓ Proved
Answer \( \ln{\left(4 x + 1 \right)} \)
Top: the graph of f = -x + (x + 1/4)·log(4x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4(x + 1/4)/(4x + 1) + log(4x + 1) - 1.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(- x + \left(x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)}\right) \]
derivative✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} \left(x + \frac{1}{4}\right) \ln{\left(4 x + 1 \right)} \]
constant-multiplesimplifyDerivative of x+1/4 is 1 Combine terms✓ Proved
\[ = \frac{\left(x + \frac{1}{4}\right) \frac{d}{d x} \left(4 x + 1\right)}{4 x + 1} + \ln{\left(4 x + 1 \right)} - 1 \]
logarithmicDerivative of log(u) is u'/u✓ Proved
\[ = \frac{4 \left(x + \frac{1}{4}\right)}{4 x + 1} + \ln{\left(4 x + 1 \right)} - 1 \]
constant-multiplesimplifyDerivative of 4*x+1 is 4 Simplify the fraction✓ Proved
\[ = \ln{\left(4 x + 1 \right)} \]
simplifysimplifysimplifysimplifyDistribute the 4 4*(x+1/4) = 4x+1 Cancel the fraction Combine -1 and +1✓ 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 log(x). Simplify the term x**2 * (2/x).✓ Proved
\[ = x \ln{\left(x \right)} \]
algebraalgebrasimplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved
Answer \( x \log{\left(x \right)} \)
Top: the graph of f = x²·(2log(x) - 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x·(2log(x) - 1)/2 + x/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
logarithmicsimplifyDerivative of 2*log(x)-1 is 2/x. Simplify (x**2/4)*(2/x) to x/2.✓ Proved
\[ = x \ln{\left(x \right)} \]
simplifyCombine terms to get the 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
derivativeconstant-multiplealgebraDifferentiate 1 and log(x). Simplify the constant factor. Simplify the term x**2 * (-2/x).✓ Proved
\[ = - x \ln{\left(x \right)} \]
algebraalgebrasimplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved
Answer \( - x \log{\left(x \right)} \)
Top: the graph of f = x²·(1 - 2log(x))/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x·(1 - 2log(x))/2 - x/2.
A second solution, written separately
By qwen3.6:35b-mlx, without seeing the first. SymPy checked every line.
logarithmicsimplifyDifferentiate log(x) to get 1/x. Simplify the term inside the parentheses.✓ Proved
\[ = - x \ln{\left(x \right)} \]
algebrasimplifysimplifyDistribute x^2 * (-2/x) to get -2x and expand the first term. Combine like terms: 2x - 2x cancels out. Divide by 4 to get the final result.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Differentiate \( \displaystyle f(x) = x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5} \).
\[ \frac{d}{d x} \left(x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} \]
sumApply the sum rule.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{x \frac{d}{d x} \left(5 x + 1\right)}{5 x + 1} + \ln{\left(5 x + 1 \right)} + \frac{\frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} - 1 \]
chainApply the chain rule to the log terms.✓ Proved
\[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} - 1 + \frac{1}{5 x + 1} \]
derivativealgebraDifferentiate the inner function 5x + 1. Simplify the terms.✓ Proved
\[ = \ln{\left(5 x + 1 \right)} \]
algebraalgebrasimplifyCombine the fractions. Simplify the fraction. Final simplification.✓ Proved
Answer \( \log{\left(5 x + 1 \right)} \)
Top: the graph of f = x·log(5x + 1) - x + log(5x + 1)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(5x + 1) + log(5x + 1) - 1 + 1/(5x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(5 x + 1 \right)} - x + \frac{\ln{\left(5 x + 1 \right)}}{5}\right) \]
derivative✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} \]
sum✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]
product✓ Proved
\[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{d}{d x} \frac{\ln{\left(5 x + 1 \right)}}{5} - 1 \]
chain✓ Proved
\[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x + \frac{\frac{d}{d x} \ln{\left(5 x + 1 \right)}}{5} - 1 \]
constant-multiple✓ Proved
\[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} \frac{d}{d x} x - 1 + \frac{1}{5 x + 1} \]
chain✓ Proved
\[ = \frac{5 x}{5 x + 1} + \ln{\left(5 x + 1 \right)} - 1 + \frac{1}{5 x + 1} \]
simplify✓ Proved
\[ = \ln{\left(5 x + 1 \right)} \]
simplifysimplifysimplify✓ 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 \( \displaystyle f(x) = x \ln{\left(5 x - 3 \right)} - x - \frac{3 \ln{\left(5 x - 3 \right)}}{5} \).
\[ \frac{d}{d x} \left(x \ln{\left(5 x - 3 \right)} - x - \frac{3 \ln{\left(5 x - 3 \right)}}{5}\right) \]
Start with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} x + \frac{d}{d x} x \ln{\left(5 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 3 \right)}}{5} \]
sumApply the sum rule for differentiation.✓ Proved
\[ = \frac{d}{d x} x \ln{\left(5 x - 3 \right)} - \frac{d}{d x} \frac{3 \ln{\left(5 x - 3 \right)}}{5} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} \frac{d}{d x} x - \frac{3 \frac{d}{d x} \ln{\left(5 x - 3 \right)}}{5} - 1 \]
productApply the product rule to the first term.✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(5 x - 3 \right)}}{5} - 1 \]
constantThe derivative of x is 1.✓ Proved
\[ = \frac{x \frac{d}{d x} \left(5 x - 3\right)}{5 x - 3} + \ln{\left(5 x - 3 \right)} - 1 - \frac{3 \frac{d}{d x} \left(5 x - 3\right)}{5 \left(5 x - 3\right)} \]
chainApply the chain rule to the log terms.✓ Proved
\[ = \frac{5 x}{5 x - 3} + \ln{\left(5 x - 3 \right)} - 1 - \frac{3}{5 x - 3} \]
derivativealgebraThe derivative of 5x - 3 is 5. Simplify the fractions.✓ Proved
\[ = \ln{\left(5 x - 3 \right)} \]
algebraalgebrasimplifyCombine the terms with the same denominator. Simplify the fraction (5x-3)/(5x-3). Final simplification.✓ Proved
Answer \( \ln{\left(5 x - 3 \right)} \)
Top: the graph of f = x·log(5x - 3) - x - 3log(5x - 3)/5. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(5x - 3) + log(5x - 3) - 1 - 3/(5x - 3).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(x \ln{\left(5 x - 3 \right)} - x - \frac{3 \ln{\left(5 x - 3 \right)}}{5}\right) \]
derivative✓ Proved
\[ = x \frac{d}{d x} \ln{\left(5 x - 3 \right)} + \ln{\left(5 x - 3 \right)} - \frac{3 \frac{d}{d x} \ln{\left(5 x - 3 \right)}}{5} - 1 \]
productApply product rule to x*log(5*x-3) and constant multiple rule to -3*log(5*x-3)/5✓ Proved
\[ = \frac{5 x}{5 x - 3} + \ln{\left(5 x - 3 \right)} - 1 - \frac{3}{5 x - 3} \]
chainsimplifyDifferentiate log(5*x-3) using chain rule Simplify 3/5*5 to 3✓ Proved
\[ = \ln{\left(5 x - 3 \right)} \]
simplifyCombine 5*x/(5*x-3) - 3/(5*x-3) to 1, cancel with -1✓ 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
derivativealgebraalgebraDifferentiate the individual components. Simplify the expression inside the parentheses. Distribute the 3 into the parentheses.✓ Proved
\[ = 3 x \frac{d}{d x} \ln{\left(x \right)} + 3 \ln{\left(x \right)} \frac{d}{d x} x - \frac{d}{d x} 3 x \]
productApply product rule to 3*x*log(x) and differentiate -3*x✓ Proved
\[ = 3 \ln{\left(x \right)} \]
derivativesimplifyalgebrasimplifyDerivative of x is 1; derivative of log(x) is 1/x Simplify x*(1/x) to 1 Distribute the 3 Combine constants 3 - 3 = 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
logarithmicalgebraDifferentiate the log(x + 1) terms. Simplify the expression.✓ Proved
\[ = \ln{\left(x + 1 \right)} \]
algebraalgebrasimplifyCombine the fractions with the same denominator. Simplify the fraction (x + 1)/(x + 1). Final simplification.✓ Proved
Answer \( \ln{\left(x + 1 \right)} \)
Top: the graph of f = x·log(x + 1) - x + log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x/(x + 1) + log(x + 1) - 1 + 1/(x + 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
logarithmicsimplifysimplifyDerivative of log(x+1) is 1/(x+1) Cancel (x+1)/(x+1) 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
logarithmicconstant-multipleDifferentiate the logarithmic terms. Simplify the coefficients.✓ Proved
\[ = \ln{\left(x + 2 \right)} \]
algebrasimplifysimplifyCombine the fractions with the same denominator. Simplify the fraction. Final simplification.✓ Proved
Answer \( \ln{\left(x + 2 \right)} \)
Top: the graph of f = x·log(x + 2) - x + 2log(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x/(x + 2) + log(x + 2) - 1 + 2/(x + 2).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
chainsimplifyDifferentiate log(x+2) using chain rule. Remove explicit multiplication by 1.✓ Proved
\[ = \ln{\left(x + 2 \right)} \]
simplifysimplifysimplifyCombine the two fractions. Simplify (x+2)/(x+2) to 1. Cancel the +1 and -1.✓ 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
algebraalgebraCombine the fractions with the same denominator. Factor out 5 from the numerator.✓ Proved
\[ = 5 \ln{\left(x - 1 \right)} \]
algebrasimplifySimplify the fraction (x - 1)/(x - 1) to 1. Combine the remaining constant terms.✓ Proved
Answer \( 5 \ln{\left(x - 1 \right)} \)
Top: the graph of f = 5x·log(x - 1) - 5x - 5log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5x/(x - 1) + 5log(x - 1) - 5 - 5/(x - 1).
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \frac{d}{d x} \left(5 x \ln{\left(x - 1 \right)} - 5 x - 5 \ln{\left(x - 1 \right)}\right) \]
derivativesimplifyDifferentiate log(x-1) and x Combine constants✓ Proved
\[ = 5 \ln{\left(x - 1 \right)} \]
simplifysimplifyCombine the two fractions Cancel the remaining 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
derivativeconstant-multipleDifferentiate each term separately. Distribute the constant 2.✓ Proved
\[ = \frac{2 x + 2}{x} \]
algebraCombine the terms into a single fraction.✓ Proved
Answer \( 2 + \frac{2}{x} \)
Top: the graph of f = log(x²·e^(2x)). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (2x²·e^(2x) + 2x·e^(2x))·e^(-2x)/x².
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
productDifferentiate the product x**2 * exp(2*x).✓ Proved
\[ = \frac{2 \left(x + 1\right)}{x} \]
algebraFactor out common terms from the numerator.✓ Proved
\[ = \frac{2 x + 2}{x} \]
simplifyCancel exp(2*x) and one x to simplify 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
productApply product rule to (x-1)*log(x-1)✓ Proved
\[ = \ln{\left(x - 1 \right)} \]
derivativealgebrasimplifyDerivative of x-1 is 1; derivative of log(x-1) is 1/(x-1) Simplify (x-1)*(1/(x-1)) to 1 Combine -1 and +1✓ 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