Derivative of \( \displaystyle - \ln{\left(\sin{\left(x + 2 \right)} \right)} \)
Problem 2.1662 · hard
Differentiate \( \displaystyle f(x) = - \ln{\left(\sin{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(\sin{\left(x + 2 \right)} \right)} \]constantMove the negative sign outside the derivative.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sin{\left(x + 2 \right)}}{\sin{\left(x + 2 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\cos{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)}{\sin{\left(x + 2 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\cos{\left(x + 2 \right)}}{\sin{\left(x + 2 \right)}} \]derivative algebraDifferentiate the inner function x + 2. Simplify the expression.✓ Proved
- \[ = - \frac{1}{\tan{\left(x + 2 \right)}} \]simplifyRewrite using the cotangent identity.✓ Proved
Answer \( - \frac{1}{\tan{\left(x + 2 \right)}} \)
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 + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x + 2) = 0 tan has poles at odd multiples of pi/2 undefined where tan(x + 2) = 0 |
| 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 + 2) = 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 and logarithmic differentiation rules in separate steps. The final simplification to -cot(x+2) expressed as -1/tan(x+2) is algebraically correct and follows the allowed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the chain rule and logarithmic differentiation rules in separate steps. The final simplification to -cot(x+2) expressed as -1/tan(x+2) is algebraically correct and follows the allowed vocabulary.qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06
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.