Derivative of \( \displaystyle \frac{5 \ln{\left(\ln{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.847 · hard Video
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\ln{\left(3 x - 3 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\ln{\left(3 x - 3 \right)} \right)}}{3} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\ln{\left(3 x - 3 \right)} \right)}}{3} \]constant-multiple rewrite simplifyPull 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
- \[ = \frac{5}{\left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \]derivative algebra simplifyDifferentiate the innermost linear function. Multiply the terms together. Cancel the 3 in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\left(3 x - 3\right) \ln{\left(3 x - 3 \right)}} \)
Mind the domain. The answer is also defined on (1, 4/3), where f(x) is not. Substituting there gives a number that is not a slope of f.
✓ 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x - 3) = 0 undefined where 3*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x - 3) = 0 undefined where 3*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x - 3) = 0 undefined where 3*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x - 3) = 0 undefined where 3*x - 3 = 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 log(3*x - 3) = 0 undefined where 3*x - 3 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 3 rewrites the inner logarithm incorrectly and applies two transformations at once. Step 9 claims to cancel a 3 that no longer exists, making the final simplification invalid.qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule and the chain rule in sequence. The intermediate rewrite and simplify steps (3 and 4) are redundant but algebraically valid and do not constitute a defect.
Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line is a correct equality and the rule labels are legitimate. The rewrite in step 3 and its reversal in step 4 are a pointless detour, and 'for clarity' is a poor description, but nothing is false and the derivative is correct.
gpt-oss:20b: dismiss — Step 3 replaces 3x-3 with exp(log(3x-3)), which is valid on the domain (3x-3>0) and is a single rewrite; step 9 does cancel a real 3, since line 8 has 5/3 * (3/...), so both claims are false.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Every line is a correct equality and the rule labels are legitimate. The rewrite in step 3 and its reversal in step 4 are a pointless detour, and 'for clarity' is a poor description, but nothing is false and the derivative is correct.qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the constant multiple rule and the chain rule in sequence. The intermediate rewrite and simplify steps (3 and 4) are redundant but algebraically valid and do not constitute a defect.gpt-oss:20b: fail (error) 2026-09-27 — Step 3 rewrites the inner logarithm incorrectly and applies two transformations at once. Step 9 claims to cancel a 3 that no longer exists, making the final simplification invalid.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (misleading) 2026-09-26 — Step 9 claims to cancel a 3 in the numerator and denominator, but after step 8 the expression is already 5/(log(3*x-3)*(3*x-3)) with no remaining factor of 3 to cancel. This misleads the reader about the algebraic simplification performed.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in a logical sequence. The intermediate rewrite and simplify steps (3 and 4) are redundant but do not constitute a defect as they are algebraically valid and do not violate the single-rule-per-step constraint.gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.