Derivative of \( \displaystyle \sin^{x}{\left(x \right)} \)
Problem 2.581 · hard Video
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 independently
- \[ = e^{x \ln{\left(\sin{\left(x \right)} \right)}} \frac{d}{d x} x \ln{\left(\sin{\left(x \right)} \right)} \]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 independently
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 by independent solution. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, 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 claude-sonnet-5-5 script, run by sympy 1.14.0: derivatives of sin(x)**x and exp(x*log(sin(x))) agree after rewriting the power as exp(x*log(sin(x))) ( checks/2.581/line2-24cdb223.py)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 independently | claude-sonnet-5-5 script, run by sympy 1.14.0, 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 claude-sonnet-5-5 script, run by sympy 1.14.0: replacing exp(x*log(sin(x))) by sin(x)**x turns line 7 into line 8 ( checks/2.581/line8-36f10683.py)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 (error) — Step 3 incorrectly applies the product rule to the exponential function. The derivative of exp(g) is exp(g)·g', which is a chain rule application, not a product rule. The label should be "chain" and the step should not be described as a product rule.qwen3.6:27b-mlx: fail (error) — Step 1 is labeled 'rewrite' but performs the differentiation operation (unfolding the derivative), which violates the 'one rule per step' constraint and mislabels the action. Step 3 is labeled 'product' but applies the chain rule to the outer exponential function, which is a mislabeling of the rule applied.
Senior review claude-sonnet-5-5, 2026-10-10: fail — The math is correct and every line is equivalent, but the rule labels on lines 2 to 4 are shifted. The sin(x)**x to exp(x*log(sin(x))) step is labeled 'chain' when it is a rewrite. The exp(g) to exp(g)*g' step is labeled 'product' when it is the chain rule. The product-rule expansion in line 4 is labeled 'derivative'. A student would learn the wrong rule names.
gpt-oss:20b: uphold — The step from line 2 to line 3, exp(g) to exp(g)*g', is the chain rule, yet line 3 is labeled 'product' with a note claiming the product rule was applied to the exponent, which is a wrong rule name.qwen3.6:27b-mlx: uphold — The objection to step 3 is correct: the outer-exponential differentiation is a chain-rule step labeled 'product'. The objection to step 1 is dismissed, since line 1 only restates the problem and makes no transformation. The mislabeling is real, though the 'two rules in one step' reasoning is wrong.(a panel model has both passed and objected to this solution under the same prompt): dismiss — No specific defect is stated, so there is nothing to uphold. The real labeling defects were identified separately.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-10 — The math is correct and every line is equivalent, but the rule labels on lines 2 to 4 are shifted. The sin(x)**x to exp(x*log(sin(x))) step is labeled 'chain' when it is a rewrite. The exp(g) to exp(g)*g' step is labeled 'product' when it is the chain rule. The product-rule expansion in line 4 is labeled 'derivative'. A student would learn the wrong rule names.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 1 is labeled 'rewrite' but performs the differentiation operation (unfolding the derivative), which violates the 'one rule per step' constraint and mislabels the action. Step 3 is labeled 'product' but applies the chain rule to the outer exponential function, which is a mislabeling of the rule applied.gpt-oss:20b: fail (error) 2026-09-26 — Step 3 incorrectly applies the product rule to the exponential function. The derivative of exp(g) is exp(g)·g', which is a chain rule application, not a product rule. The label should be "chain" and the step should not be described as a product rule.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.