Derivative of \( \displaystyle \frac{\ln{\left(5 x \right)}}{25 x^{2}} \)
Problem 2.499 · hard Video
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 \( \frac{1 - 2 \log{\left(5 x \right)}}{25 x^{3}} \)
✓ 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. How we verify
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: passqwen3.6:27b-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.
Senior review claude-sonnet-5-5, 2026-10-09: pass — The math is correct at every line and SymPy proved each one. Steps 7-9 expand the log and then recombine it, which adds nothing, but they are valid and not misleading.
qwen3.6:27b-mlx: dismiss — Step 3 evaluates the two pending derivatives (1/x via chain, 50x via power) in a single pass, which counts as one derivative-evaluation step, and 'chain' names a rule that was actually applied, so there is no wrong label or independent double change.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-09 — The math is correct at every line and SymPy proved each one. Steps 7-9 expand the log and then recombine it, which adds nothing, but they are valid and not misleading.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 2026-09-21qwen3.6:27b-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.