Derivative of \( \displaystyle 2 \ln{\left(\sin{\left(x - 3 \right)} \right)} \)
Problem 2.1927 · hard
Differentiate \( \displaystyle f(x) = 2 \ln{\left(\sin{\left(x - 3 \right)} \right)} \).
- \[ \frac{d}{d x} 2 \ln{\left(\sin{\left(x - 3 \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 \frac{d}{d x} \ln{\left(\sin{\left(x - 3 \right)} \right)} \]constant-multiplePull the constant out of the derivative.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \sin{\left(x - 3 \right)}}{\sin{\left(x - 3 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{2 \cos{\left(x - 3 \right)}}{\sin{\left(x - 3 \right)}} \]derivative algebraDifferentiate the inner function sin(x - 3). Combine the terms into a single fraction.✓ Proved
- \[ = 2 \cot{\left(x - 3 \right)} \]simplifyUse the identity cot(u) = cos(u)/sin(u).✓ Proved
Answer \( \frac{2}{\tan{\left(x - 3 \right)}} \)
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.
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x - 3) = 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 - 3) = 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: fail (style) — The final answer is given as 2/tan(x - 3), but the last step of the derivation is 2*cot(x - 3). These are not identical expressions in the provided steps; a final step converting cot to 1/tan is missing or the stated answer does not match the derivation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (style) 2026-10-09 — The final answer is given as 2/tan(x - 3), but the last step of the derivation is 2*cot(x - 3). These are not identical expressions in the provided steps; a final step converting cot to 1/tan is missing or the stated answer does not match the derivation.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "style", "notes": "Step 3 applies the chain rule to the logarithm but also performs the differentiation of the outer function (log) into 1/u in a single step. The contgpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.