Derivative of \( \displaystyle 5 \ln{\left(\sin{\left(x \right)} \right)} \)
Problem 2.590 · hard
Differentiate \( \displaystyle f(x) = 5 \ln{\left(\sin{\left(x \right)} \right)} \).
- \[ \frac{d}{d x} 5 \ln{\left(\sin{\left(x \right)} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} \ln{\left(\sin{\left(x \right)} \right)} \]constant-multiplePull the constant out of the derivative.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{5 \cos{\left(x \right)}}{\sin{\left(x \right)}} \]derivativeDifferentiate the inner function sin(x).✓ Proved
- \[ = 5 \cot{\left(x \right)} \]simplifySimplify the expression using the cotangent identity.✓ Proved
Answer \( \frac{5}{\tan{\left(x \right)}} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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 tan has poles at odd multiples of pi/2 undefined where tan(x) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 1 lacks a proper rule label; it should be labeled "derivative". All other steps correctly apply a single rule from the allowed vocabulary.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 1 lacks a proper rule label; it should be labeled "derivative". All other steps correctly apply a single rule from the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.