Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(5 x - 1 \right)} \right)}}{5} \)
Problem 2.1975 · 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
- \[ = - \frac{\cos{\left(5 x - 1 \right)} \frac{d}{d x} \left(5 x - 1\right)}{5 \sin{\left(5 x - 1 \right)}} \]trigApply the chain rule for the sine function.✓ Proved
- \[ = - \frac{\cos{\left(5 x - 1 \right)}}{\sin{\left(5 x - 1 \right)}} \]derivative algebra algebraDifferentiate the inner linear function. Cancel the constant 5 with the denominator 5. 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-10gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (style) 2026-10-10 — Step 3 is labeled 'logarithmic' but applies the chain rule to the logarithm; the label should be 'chain' (or 'logarithmic' if it implies the derivative of log, but the step explicitly shows the chain rule structure, making 'chain' the more precise label for the operation performed on the composite function, though 'logarithmic' is acceptable if it refers to the derivative of the outer function. However, Step 4 is labeled 'trig' but applies the chain rule to the sine function. The label 'trig' is not in the allowed vocabulary for differentiation rules (only 'derivative', 'chain', etc. are). The step applies the chain rule, so it should be labeled 'chain'. Furthermore, Step 3 applies the derivative of log AND the chain rule structure in one go, which might be considered two rules, but the main issue is Step 4's label 'trig' is not a valid differentiation rule label in the provided list for this context (it's a 'chain' step). Actually, looking at the vocabulary: 'trig' is in the list. But 'trig' usually implies simplifying trig expressions or basic trig derivatives. Step 4 is `Derivative(sin(u), x) -> cos(u)*u'`. This is the chain rule. Labeling it 'trig' is misleading because it hides the chain rule application. The label should be 'chain'. Step 3 is also a chain rule application. Labeling it 'logarithmic' is acceptable if it means 'derivative of log', but the step shows the full chain rule expansion. Let's look closer. Step 3: `Derivative(log(sin(...)), x)` becomes `(1/sin(...)) * Derivative(sin(...), x)`. This is the chain rule. The label 'logarithmic' is weak but perhaps acceptable as 'derivative of log'. Step 4: `Derivative(sin(...), x)` becomes `cos(...) * Derivative(..., x)`. This is the chain rule. The label 'trig' is incorrect because it suggests a trigonometric identity or basic derivative without chain, but the chain rule is the operative principle here. The vocabulary has 'chain'. So Step 4 should be 'chain'. Step 3 should also be 'chain'. Using 'logarithmic' and 'trig' for chain rule steps is a defect in labeling.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.