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

Problem 2.1 · medium Mental math

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

\[ \frac{d}{d x} \ln{\left(\ln{\left(x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(x \right)}}{\ln{\left(x \right)}} \]
chainApply the chain rule to the outer logarithm.✓ Proved
\[ = \frac{1}{x \ln{\left(x \right)}} \]
derivative simplifyDifferentiate the inner function log(x). Simplify the product of the fractions.✓ Proved

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

Mind the domain. The answer is also defined on (0, 1), 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 undefined where log(x) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where log(x) = 0 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 undefined where log(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(x) = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

deepseek-r1:70b: pass
gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (14)

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 chain rule and standard derivative rules in separate steps, adhering to the one-change-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
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 — The solution correctly applies the chain rule and derivative rules in separate steps, adhering to the one-change-per-step constraint. The labels and notes are accurate.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-16

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.