Home›Calculus 1›Derivatives of logarithms›Problem 2.499

Derivative of \( \displaystyle \frac{\ln{\left(5 x \right)}}{25 x^{2}} \)

Problem 2.499 · hard

Differentiate \( \displaystyle f(x) = \frac{\ln{\left(5 x \right)}}{25 x^{2}} \).

\[ \frac{d}{d x} \frac{\ln{\left(5 x \right)}}{25 x^{2}} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{25 x^{2} \frac{d}{d x} \ln{\left(5 x \right)} - \ln{\left(5 x \right)} \frac{d}{d x} 25 x^{2}}{625 x^{4}} \]
quotientApply the quotient rule.✓ Proved
\[ = \frac{- 50 x \ln{\left(5 x \right)} + 25 x}{625 x^{4}} \]
chain algebraDifferentiate the numerator and denominator. Simplify the numerator term (1/x)*25*x**2.✓ Proved
\[ = \frac{1 - 2 \ln{\left(5 x \right)}}{25 x^{3}} \]
algebra algebraFactor out 25*x from the numerator. Simplify the fraction by canceling 25*x.✓ Proved
\[ = \frac{- 2 \ln{\left(x \right)} - 2 \ln{\left(5 \right)} + 1}{25 x^{3}} \]
algebra algebraExpand the logarithm. Distribute the -2.✓ Proved
\[ = \frac{1 - 2 \ln{\left(5 x \right)}}{25 x^{3}} \]
simplifyRecombine the logarithms.✓ Proved

Answer \( \frac{1 - 2 \log{\left(5 x \right)}}{25 x^{3}} \)

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. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain', but it actually applies the derivative rule for log(5*x) (which involves the chain rule) AND the derivative rule for 25*x**2 (power rule) simultaneously. The contract requires each step to change only one thing. Additionally, the note 'Differentiate the numerator and denominator' implies two distinct differentiation actions were taken in one step, violating the single-change constraint.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'chain', but it actually applies the derivative rule for log(5*x) (which involves the chain rule) AND the derivative rule for 25*x**2 (power rule) simultaneously. The contract requires each step to change only one thing. Additionally, the note 'Differentiate the numerator and denominator' implies two distinct differentiation actions were taken in one step, violating the single-change constraint.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the chain rule to differentiate log(5*x) and the power rule to differentiate 25*x**2 simultaneously, violating the one-rule-per-step constraint. Additionally, Step 7 and Step 9 are redundant and logically circular, expanding and then immediately recombining the logarithm without necessity.
gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies two derivative operations (for the numerator and the denominator) in a single line, violating the rule that each step must change only one thing. It should be split into two separate derivative steps.
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 applies the chain rule to differentiate log(5*x) and the power rule to differentiate 25*x**2 simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'chain' is incorrect for the differentiation of the polynomial term 25*x**2.
gpt-oss:20b: pass 2026-09-20

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.