Derivative of \( \displaystyle \frac{5 \ln{\left(\ln{\left(3 x - 3 \right)} \right)}}{3} \)
Problem 2.847 · hard
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.
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 |
| 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 (misleading) — 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
Every verdict on record (4)
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 2026-09-26 with SymPy 1.14.0.