Derivative of \( \displaystyle \sin^{x}{\left(x \right)} \)
Problem 2.581 · hard
Differentiate \( \displaystyle f(x) = \sin^{x}{\left(x \right)} \).
- \[ \frac{d}{d x} \sin^{x}{\left(x \right)} \]rewriteRewrite the function using the exponential identity.✓ Proved
- \[ = \frac{d}{d x} e^{x \ln{\left(\sin{\left(x \right)} \right)}} \]chainApply the chain rule.≈ Checked numerically
- \[ = e^{x \ln{\left(\sin{\left(x \right)} \right)}} \frac{d}{d x} x \ln{\left(\sin{\left(x \right)} \right)} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(x \frac{d}{d x} \ln{\left(\sin{\left(x \right)} \right)} + \ln{\left(\sin{\left(x \right)} \right)} \frac{d}{d x} x\right) e^{x \ln{\left(\sin{\left(x \right)} \right)}} \]derivativeDifferentiate the terms in the product.✓ Proved
- \[ = \left(\frac{x \frac{d}{d x} \sin{\left(x \right)}}{\sin{\left(x \right)}} + \ln{\left(\sin{\left(x \right)} \right)}\right) e^{x \ln{\left(\sin{\left(x \right)} \right)}} \]derivativeDifferentiate the logarithm term using the chain rule.✓ Proved
- \[ = \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \ln{\left(\sin{\left(x \right)} \right)}\right) e^{x \ln{\left(\sin{\left(x \right)} \right)}} \]trigDifferentiate the sine function.✓ Proved
- \[ = \left(x \cot{\left(x \right)} + \ln{\left(\sin{\left(x \right)} \right)}\right) e^{x \ln{\left(\sin{\left(x \right)} \right)}} \]simplifySubstitute cot(x) for cos(x)/sin(x).✓ Proved
- \[ = \left(x \cot{\left(x \right)} + \ln{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} \]rewriteConvert the exponential back to the original base form.≈ Checked numerically
Answer \( \left(\frac{x}{\tan{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} \)
Lines: 7 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -x*exp(x*log(sin(x)))/tan(x) + x*sin(x)**x/tan(x) - exp(x*log(sin(x)))*log(sin(x)) + log(sin(x))*sin(x)**x; numeric agreement only, at 24 of 24 sampled points 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x) = 0 cot has poles at multiples of pi |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -(x/tan(x) + log(sin(x)))*sin(x)**x + (x*cot(x) + log(sin(x)))*exp(x*log(sin(x))); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments 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 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(x) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — Step 3 applies the chain rule, not the product rule; labeling it "product" is a mis‑labeling defect.qwen3.6:27b-mlx: pass — The solution correctly applies logarithmic differentiation. Each step isolates a single rule application, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies logarithmic differentiation. Each step isolates a single rule application, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 applies the chain rule, not the product rule; labeling it "product" is a mis‑labeling defect.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 is labeled 'chain' but performs no differentiation; it merely rewrites the expression, which should be labeled 'rewrite'. Step 5 is labeled 'derivative' but applies the chain rule to differentiate log(sin(x)), so it should be labeled 'chain'.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as a product rule; it also repeats the same expression without applying any new rule.
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 2026-09-26 with SymPy 1.14.0.