Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1578

Derivative of \( \displaystyle \ln{\left(\sin{\left(5 x - 1 \right)} \right)} \)

Problem 2.1578 · hard

Differentiate \( \displaystyle f(x) = \ln{\left(\sin{\left(5 x - 1 \right)} \right)} \).

\[ \frac{d}{d x} \ln{\left(\sin{\left(5 x - 1 \right)} \right)} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sin{\left(5 x - 1 \right)}}{\sin{\left(5 x - 1 \right)}} \]
chainApply the chain rule for the logarithm.✓ Proved
\[ = \frac{\cos{\left(5 x - 1 \right)} \frac{d}{d x} \left(5 x - 1\right)}{\sin{\left(5 x - 1 \right)}} \]
chainApply the chain rule for the sine function.✓ Proved
\[ = \frac{5 \cos{\left(5 x - 1 \right)}}{\sin{\left(5 x - 1 \right)}} \]
derivative algebraDifferentiate the linear inner function. Rearrange the terms.✓ Proved
\[ = 5 \cot{\left(5 x - 1 \right)} \]
simplifyUse the identity cot(u) = cos(u)/sin(u).✓ Proved

Answer \( \frac{5}{\tan{\left(5 x - 1 \right)}} \)

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.

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. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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(5*x - 1) = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 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 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule in successive steps, differentiates the inner linear function, and simplifies the result using standard trigonometric identities. The final answer is algebraically equivalent to the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the chain rule in successive steps, differentiates the inner linear function, and simplifies the result using standard trigonometric identities. The final answer is algebraically equivalent to the stated answer.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — The stated answer is 5/tan(5*x - 1), but the final step of the solution is 5 * cot(5*x - 1). While these are mathematically equivalent, the solution does not match the stated answer, and the final step should ideally simplify to the stated form or the stated answer should match the derived form. More critically, the prompt asks to review the solution against the contract. The solution ends with cot, but the stated answer is 1/tan. This is a mismatch. However, looking closer at the rules: Step 2 applies chain rule to log. Step 3 applies chain rule to sin. Step 4 applies derivative to linear term. Step 5 is algebra. Step 6 is simplify. The steps are logically sound and follow the one-rule-per-step constraint. The only issue is the mismatch between the final derived expression and the 'Stated answer'. Is this a defect in the solution steps? The solution steps are correct. The 'Stated answer' is just a label. Usually, the 'Stated answer' is what the solver claims to have found. If the solver claims the answer is 5/tan(...) but derives 5*cot(...), they haven't actually derived the stated answer. But 5*cot(...) is a valid simplification. Let's look for rule violations. Step 2: chain rule on log. Correct. Step 3: chain rule on sin. Correct. Step 4: derivative of 5x-1. Correct. Step 5: algebra rearrangement. Correct. Step 6: simplify cos/sin to cot. Correct. The mismatch between the final line and the stated answer is a consistency error in the problem statement/solution pair, but is it a defect in the *steps*? The instructions say 'judge the solution against that contract'. The contract includes the steps. The steps are correct. The stated answer is just metadata. However, often 'Stated answer' is the target. If the target is 5/tan, and the solution stops at 5*cot, it's incomplete relative to the target form, but mathematically correct. Let's look for a more subtle error. Step 2 note says 'Apply the chain rule for the logarithm'. Step 3 note says 'Apply the chain rule for the sine function'. This is correct. Step 4 note says 'Differentiate the linear inner function'. Correct. Step 6 note says 'Use the identity cot(u) = cos(u)/sin(u)'. Correct. Wait, look at Step 2. `Derivative(log(sin(5*x - 1)), x)` becomes `1/sin(5*x - 1) * Derivative(sin(5*x - 1), x)`. This is the chain rule. The rule label is `chain`. This is correct. Look at Step 3. `1/sin(5*x - 1) * Derivative(sin(5*x - 1), x)` becomes `1/sin(5*x - 1) * cos(5*x - 1) * Derivative(5*x - 1, x)`. This is the chain rule applied to the sine term. The rule label is `chain`. This is correct. Look at Step 4. `... * Derivative(5*x - 1, x)` becomes `... * 5`. The rule label is `derivative`. This is correct. Look at Step 5. `1/sin(5*x - 1) * cos(5*x - 1) * 5` becomes `5 * cos(5*x - 1) / sin(5*x - 1)`. The rule label is `algebra`. This is correct. Look at Step 6. `5 * cos(5*x - 1) / sin(5*x - 1)` becomes `5 * cot(5*x - 1)`. The rule label is `simplify`. This is correct. The only potential issue is the mismatch between the final result `5 * cot(5*x - 1)` and the `Stated answer: 5/tan(5*x - 1)`. In many contexts, `cot` and `1/tan` are considered equivalent forms. However, if the task was to differentiate and the stated answer is a specific form, the solution should ideally reach that form. But is it a *defect* in the steps? No. The steps are valid. Let's re-read the prompt carefully. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So the math is fine. 'Look for: a step that applies more than one rule at once, a label that names a rule the step did NOT apply, a note that would teach a student something false'. Is there a note that teaches something false? Step 2 note: 'Apply the chain rule for the logarithm.' This is slightly imprecise language (you apply the chain rule to the composition), but not false. Step 6 note: 'Use the identity cot(u) = cos(u)/sin(u).' This is true. Is there a label error? Step 2: chain. Correct. Step 3: chain. Correct. Step 4: derivative. Correct. Step 5: algebra. Correct. Step 6: simplify. Correct. Is there a step that applies two rules? Step 2: Only chain rule. Step 3: Only chain rule. Step 4: Only derivative. Step 5: Only algebra (rearranging multiplication/division). Step 6: Only simplify (trig identity). It seems the solution is perfect. The mismatch with the stated answer is likely just an alternative form. `5/tan` and `5*cot` are the same. The solution is sound.
gpt-oss:20b: fail (style) 2026-10-05 — Step 1 lacks a proper rule label; it should be marked as a derivative step. All other steps correctly apply single rules and the final simplification is valid.

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 2026-10-05 with SymPy 1.14.0.