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

Problem 2.21 · hard

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

\[ \frac{d}{d x} \ln{\left(\sin{\left(x \right)} \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \]
derivative algebraDifferentiate the inner function sin(x). Combine the fraction.✓ Proved
\[ = \cot{\left(x \right)} \]
simplifySimplify the ratio of cosine to sine.✓ Proved

Answer \( \cot{\left(x \right)} \)

Mind the domain. The answer is also defined at points 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 sin(x) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0 cot has poles at multiples of pi
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 cot has poles at multiples of pi
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 — The solution correctly applies the chain rule and standard derivatives, with each step changing only one aspect of the expression and using valid labels from the fixed vocabulary.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and standard derivatives, with each step changing only one aspect of the expression and using valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule and standard derivatives, with each step changing only one aspect of the expression. The labels used are from the allowed vocabulary and accurately describe the operations performed.
gpt-oss:20b: pass 2026-09-20
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.