Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \)
Problem 2.1677 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x + 1 \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sin{\left(2 x + 1 \right)}}{2 \sin{\left(2 x + 1 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\cos{\left(2 x + 1 \right)} \frac{d}{d x} \left(2 x + 1\right)}{2 \sin{\left(2 x + 1 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\cos{\left(2 x + 1 \right)}}{\sin{\left(2 x + 1 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the expression by canceling the 2 and the 1/2.✓ Proved
- \[ = - \cot{\left(2 x + 1 \right)} \]simplifyUse the trigonometric identity cot(u) = cos(u)/sin(u).✓ Proved
Answer \( - \frac{1}{\tan{\left(2 x + 1 \right)}} \)
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 |
| 3 | ✓ 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 |
| 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 |
| 7 | ✓ 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (style) 2026-10-06 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, which is a defect in granularity/naming. Step 3 is labeled 'logarithmic' but applies the chain rule (derivative of outer log times derivative of inner), so the label is misleading.
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 2026-10-06 with SymPy 1.14.0.