Derivative of \( \displaystyle \frac{5 \ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \)
Problem 2.1601 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\sin{\left(2 x - 3 \right)} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sin{\left(2 x - 3 \right)}}{2 \sin{\left(2 x - 3 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{5 \cos{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \sin{\left(2 x - 3 \right)}} \]chainApply the chain rule for the sine function.✓ Proved
- \[ = \frac{5 \cos{\left(2 x - 3 \right)}}{\sin{\left(2 x - 3 \right)}} \]derivative algebraDifferentiate the inner linear function. Simplify the constants.✓ Proved
- \[ = 5 \cot{\left(2 x - 3 \right)} \]simplifyUse the cotangent identity.✓ Proved
Answer \( \frac{5}{\tan{\left(2 x - 3 \right)}} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(2*x - 3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(2*x - 3) = 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 - 3) = 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: fail (style) — Step 3 is labeled 'chain', but the transformation from Step 2 to Step 3 is the application of the logarithmic derivative rule (d/dx log(u) = u'/u), which should be labeled 'logarithmic'. The chain rule is applied in the subsequent step to evaluate u'. Using 'chain' here conflates the outer function's derivative rule with the inner function's differentiation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-05 — Step 3 is labeled 'chain', but the transformation from Step 2 to Step 3 is the application of the logarithmic derivative rule (d/dx log(u) = u'/u), which should be labeled 'logarithmic'. The chain rule is applied in the subsequent step to evaluate u'. Using 'chain' here conflates the outer function's derivative rule with the inner function's differentiation.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — Step 3 is labeled 'chain' but applies the derivative of the logarithm (logarithmic rule); the chain rule is not applied until the inner function is differentiated in subsequent steps. This conflates the outer derivative rule with the chain rule structure.gpt-oss:20b: pass 2026-10-05
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.