Derivative of \( \displaystyle \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \)
Problem 2.445 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\ln{\left(3 x \right)} \right)}}{3} \]chainApply the chain rule to the outer logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(3 x \right)}}{3 \ln{\left(3 x \right)}} \]chainApply the chain rule to the inner logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 3 x}{9 x \ln{\left(3 x \right)}} \]derivativeDifferentiate the innermost function 3*x.✓ Proved
- \[ = \frac{1}{3 x \ln{\left(3 x \right)}} \]algebra algebra simplifySimplify the constant term. Cancel the 3 in the numerator and denominator. Final simplified expression.✓ Proved
Answer \( \frac{1}{3 x \log{\left(3 x \right)}} \)
Mind the domain. The answer is also defined on (0, 1/3), 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.
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 undefined where log(3*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(3*x) = 0 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 log(3*x) = 0 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 log(3*x) = 0 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 log(3*x) = 0 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 log(3*x) = 0 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: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, followed by successive applications of the chain rule, and finishes with algebraic simplification. Each step isolates a single transformation and uses an appropriate label from the fixed vocabulary.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.