Derivative of \( \displaystyle \sin^{4 x}{\left(4 x \right)} \)
Problem 2.602 · hard
Differentiate \( \displaystyle f(x) = \sin^{4 x}{\left(4 x \right)} \).
- \[ \frac{d}{d x} \sin^{4 x}{\left(4 x \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \]rewriteRewrite the power using the exponential and logarithm.≈ Checked numerically
- \[ = e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \frac{d}{d x} 4 x \ln{\left(\sin{\left(4 x \right)} \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(4 x \frac{d}{d x} \ln{\left(\sin{\left(4 x \right)} \right)} + \ln{\left(\sin{\left(4 x \right)} \right)} \frac{d}{d x} 4 x\right) e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(4 x \frac{d}{d x} \ln{\left(\sin{\left(4 x \right)} \right)} + 4 \ln{\left(\sin{\left(4 x \right)} \right)}\right) e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \]constant-multipleDifferentiate 4*x.✓ Proved
- \[ = \left(\frac{4 x \frac{d}{d x} \sin{\left(4 x \right)}}{\sin{\left(4 x \right)}} + 4 \ln{\left(\sin{\left(4 x \right)} \right)}\right) e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = \left(\frac{16 x \cos{\left(4 x \right)}}{\sin{\left(4 x \right)}} + 4 \ln{\left(\sin{\left(4 x \right)} \right)}\right) e^{4 x \ln{\left(\sin{\left(4 x \right)} \right)}} \]trig algebraDifferentiate sin(4*x). Multiply the constants.✓ Proved
- \[ = \left(16 x \cot{\left(4 x \right)} + 4 \ln{\left(\sin{\left(4 x \right)} \right)}\right) \sin^{4 x}{\left(4 x \right)} \]simplifySubstitute back the original base and simplify the fraction.≈ Checked numerically
Answer \( \left(\frac{16 x}{\tan{\left(4 x \right)}} + 4 \log{\left(\sin{\left(4 x \right)} \right)}\right) \sin^{4 x}{\left(4 x \right)} \)
✓ Nihil obstat Lines: 8 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 4*(4*x*cos(4*x) + log(sin(4*x))*sin(4*x))*(-exp(4*x*log(sin(4*x))) + sin(4*x)**(4*x))/sin(4*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(4*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(4*x) = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((4*x*cos(4*x) + log(sin(4*x))*sin(4*x))*exp(4*x*log(sin(4*x))) - (4*x*cot(4*x) + log(sin(4*x)))*sin(4*x)**(4*x + 1))/sin(4*x); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(4*x) = 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 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(4*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 logarithmic differentiation, adhering to the one-rule-per-step constraint and using valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies logarithmic differentiation, adhering to the one-rule-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 is labeled 'constant-multiple' but performs the differentiation of 4*x (a 'derivative' step). Step 9 is labeled 'simplify' but performs two distinct operations: substituting the exponential form back to power form and converting cos/sin to cot (which should be separate steps or labeled 'rewrite').gpt-oss:20b: pass 2026-09-21
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.