Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(5 x + 1 \right)} \right)}}{5} \)
Problem 2.2047 · hard
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\sin{\left(5 x + 1 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(5 x + 1 \right)} \right)}}{5}\right) \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x + 1 \right)} \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sin{\left(5 x + 1 \right)}}{5 \sin{\left(5 x + 1 \right)}} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = - \frac{\cos{\left(5 x + 1 \right)} \frac{d}{d x} \left(5 x + 1\right)}{5 \sin{\left(5 x + 1 \right)}} \]chainApply the chain rule to the sine function.✓ Proved
- \[ = - \frac{\cos{\left(5 x + 1 \right)}}{\sin{\left(5 x + 1 \right)}} \]derivative constant-multiple algebraDifferentiate the inner linear function. Simplify the constant factors. Combine the terms into a single fraction.✓ Proved
- \[ = - \cot{\left(5 x + 1 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( - \frac{1}{\tan{\left(5 x + 1 \right)}} \)
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(5*x + 1) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x + 1) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x + 1) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x + 1) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x + 1) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*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(5*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 constant multiple rule, chain rule, and standard derivatives in a step-by-step manner. The final simplification to -cot(5*x + 1) is algebraically equivalent to the stated answer -1/tan(5*x + 1).
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the constant multiple rule, chain rule, and standard derivatives in a step-by-step manner. The final simplification to -cot(5*x + 1) is algebraically equivalent to the stated answer -1/tan(5*x + 1).gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (style) 2026-10-11 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 6 is labeled 'constant-multiple' but performs arithmetic simplification of constants, which should be labeled 'algebra' or 'simplify'.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.