Derivative of \( \displaystyle \sin^{x + 2}{\left(x + 2 \right)} \)
Problem 2.357 · hard
Differentiate \( \displaystyle f(x) = \sin^{x + 2}{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \sin^{x + 2}{\left(x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \]rewriteRewrite the base and exponent using the exponential identity.≈ Checked numerically
- \[ = e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \frac{d}{d x} \left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)} \]chainApply the chain rule for the exponential function.✓ Proved
- \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\sin{\left(x + 2 \right)} \right)} + \ln{\left(\sin{\left(x + 2 \right)} \right)} \frac{d}{d x} \left(x + 2\right)\right) e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \]productApply the product rule to the exponent's derivative.✓ Proved
- \[ = \left(\left(x + 2\right) \frac{d}{d x} \ln{\left(\sin{\left(x + 2 \right)} \right)} + \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \left(\frac{\left(x + 2\right) \frac{d}{d x} \sin{\left(x + 2 \right)}}{\sin{\left(x + 2 \right)}} + \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \]derivativeApply the chain rule to the logarithm.✓ Proved
- \[ = \left(\frac{\left(x + 2\right) \cos{\left(x + 2 \right)}}{\sin{\left(x + 2 \right)}} + \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) e^{\left(x + 2\right) \ln{\left(\sin{\left(x + 2 \right)} \right)}} \]derivativeDifferentiate the sine function.✓ Proved
- \[ = \left(\frac{\left(x + 2\right) \cos{\left(x + 2 \right)}}{\sin{\left(x + 2 \right)}} + \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) \sin^{x + 2}{\left(x + 2 \right)} \]algebraConvert the exponential form back to the power form.≈ Checked numerically
- \[ = \left(\left(x + 2\right) \cot{\left(x + 2 \right)} + \ln{\left(\sin{\left(x + 2 \right)} \right)}\right) \sin^{x + 2}{\left(x + 2 \right)} \]simplifySimplify the expression using the cotangent identity.≈ Checked numerically
Answer \( \left(\frac{x + 2}{\tan{\left(x + 2 \right)}} + \log{\left(\sin{\left(x + 2 \right)} \right)}\right) \sin^{x + 2}{\left(x + 2 \right)} \)
✓ Nihil obstat Lines: 7 proved, 3 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 ((x + 2)*cos(x + 2) + log(sin(x + 2))*sin(x + 2))*(-exp((x + 2)*log(sin(x + 2))) + sin(x + 2)**(x + 2))/sin(x + 2); 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(x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(x + 2) = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left ((x + 2)*cos(x + 2) + log(sin(x + 2))*sin(x + 2))*(exp((x + 2)*log(sin(x + 2))) - sin(x + 2)**(x + 2))/sin(x + 2); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(x + 2) = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (((x + 2)*cos(x + 2) + log(sin(x + 2))*sin(x + 2))*sin(x + 2)**(x + 2) - ((x + 2)*cot(x + 2) + log(sin(x + 2)))*sin(x + 2)**(x + 3))/sin(x + 2); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where sin(x + 2) = 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(x + 2) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (10)
qwen3.6:27b-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. Each step isolates a single transformation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule, product rule, and standard derivatives in a logical sequence. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19
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.