Derivative of \( \displaystyle \sin^{5 x}{\left(5 x \right)} \)
Problem 2.425 · hard
Differentiate \( \displaystyle f(x) = \sin^{5 x}{\left(5 x \right)} \).
- \[ \frac{d}{d x} \sin^{5 x}{\left(5 x \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]rewriteRewrite the power using the exponential identity.≈ Checked numerically
- \[ = e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \frac{d}{d x} 5 x \ln{\left(\sin{\left(5 x \right)} \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(5 x \frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} \right)} + \ln{\left(\sin{\left(5 x \right)} \right)} \frac{d}{d x} 5 x\right) e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(5 x \frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} \right)} + 5 \ln{\left(\sin{\left(5 x \right)} \right)}\right) e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]derivativeDifferentiate 5*x.✓ Proved
- \[ = \left(\frac{5 x \frac{d}{d x} \sin{\left(5 x \right)}}{\sin{\left(5 x \right)}} + 5 \ln{\left(\sin{\left(5 x \right)} \right)}\right) e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{25 x \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + 5 \ln{\left(\sin{\left(5 x \right)} \right)}\right) e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]derivative algebraDifferentiate sin(5*x). Simplify the expression inside the parenthesis.✓ Proved
- \[ = \left(25 x \cot{\left(5 x \right)} + 5 \ln{\left(\sin{\left(5 x \right)} \right)}\right) e^{5 x \ln{\left(\sin{\left(5 x \right)} \right)}} \]simplifyUse the cotangent identity.✓ Proved
- \[ = 5 \left(5 x \cot{\left(5 x \right)} + \ln{\left(\sin{\left(5 x \right)} \right)}\right) \sin^{5 x}{\left(5 x \right)} \]simplifyFactor out the common 5 and rewrite the exponential base.≈ Checked numerically
Answer \( 5 \left(\frac{5 x}{\tan{\left(5 x \right)}} + \log{\left(\sin{\left(5 x \right)} \right)}\right) \sin^{5 x}{\left(5 x \right)} \)
✓ Nihil obstat Lines: 9 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -25*x*exp(5*x*log(sin(5*x)))/tan(5*x) + 25*x*sin(5*x)**(5*x)/tan(5*x) - 5*exp(5*x*log(sin(5*x)))*log(sin(5*x)) + 5*log(sin(5*x))*sin(5*x)**(5*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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) = 0 cot has poles at multiples of pi |
| 10 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -5*(5*x/tan(5*x) + log(sin(5*x)))*sin(5*x)**(5*x) + 5*(5*x*cot(5*x) + log(sin(5*x)))*exp(5*x*log(sin(5*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(5*x) = 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 — The solution correctly applies the chain rule, product rule, and standard derivatives in a logical sequence. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule, product rule, and standard derivatives in a logical sequence. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and standard derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20
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.