Home›Calculus 1›Chain rule›Problem 2.152

Derivative of \( \displaystyle - \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \)

Problem 2.152 · medium Video

Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3} \).

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\ln{\left(3 x \right)} \right)}}{3}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\ln{\left(3 x \right)} \right)}}{3} \]
constantPull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(3 x \right)}}{3 \ln{\left(3 x \right)}} \]
chainApply the chain rule to the outer logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} 3 x}{9 x \ln{\left(3 x \right)}} \]
chainApply the chain rule to the inner logarithm.✓ Proved
\[ = - \frac{1}{3 x \ln{\left(3 x \right)}} \]
derivative algebra simplifyDifferentiate the innermost linear function. Rearrange the terms for clarity. Simplify the expression by canceling the 3.✓ Proved

Answer \( - \frac{1}{3 x \log{\left(3 x \right)}} \)

Top: the graph of f = -log(log(3x))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -1/(3x·log(3x)).

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. 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 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 x = 0 undefined where log(3*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 undefined where log(3*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 undefined where log(3*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 undefined where log(3*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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-05: pass — The derivative is correct: -1/3 · 1/log(3x) · 1/(3x) · 3 = -1/(3x·log(3x)). Every step changes one thing and its rule label is legitimate, and SymPy proved each line. No specific panel objections were listed.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-05 — The derivative is correct: -1/3 · 1/log(3x) · 1/(3x) · 3 = -1/(3x·log(3x)). Every step changes one thing and its rule label is legitimate, and SymPy proved each line. No specific panel objections were listed.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, the chain rule for nested logarithms, and basic algebraic simplification. Each step isolates a single transformation and uses valid labels from the provided vocabulary.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 6 incorrectly combines the factors: -1/3 * (1/log(3*x)) * (1/(3*x)) * 3 simplifies to -1/(x*log(3*x)), not -1/3 * (1/(3*x*log(3*x))) * 3. The subsequent simplification in step 7 is therefore based on a wrong intermediate expression.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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.