Derivative of \( \displaystyle \ln{\left(\sin{\left(2 x - 1 \right)} \right)} \)
Problem 2.440 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(2 x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\sin{\left(2 x - 1 \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\cos{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{\sin{\left(2 x - 1 \right)}} \]chainApply the chain rule for the sine function.✓ Proved
- \[ = \frac{2 \cos{\left(2 x - 1 \right)}}{\sin{\left(2 x - 1 \right)}} \]derivative algebraDifferentiate the linear function inside. Combine the terms into a single fraction.✓ Proved
- \[ = 2 \cot{\left(2 x - 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{2}{\tan{\left(2 x - 1 \right)}} \)
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(2*x - 1) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 1) = 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(2*x - 1) = 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: pass — The solution correctly applies the chain rule in successive steps, uses appropriate labels from the fixed vocabulary, and arrives at the correct simplified result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule in successive steps, uses appropriate labels from the fixed vocabulary, and arrives at the correct simplified result.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule in successive steps, differentiates the inner linear function, and simplifies the result using trigonometric identities. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — The final answer is given as 2/tan(2*x - 1), but the last step of the derivation simplifies to 2*cot(2*x - 1). While algebraically equivalent, the solution does not explicitly show the step converting cot to 1/tan to match the stated answer, creating a disconnect between the derived form and the stated answer.gpt-oss:20b: pass 2026-09-20
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.