Derivative of \( \displaystyle \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \)
Problem 2.331 · hard
Differentiate \( \displaystyle f(x) = \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \).
- \[ \frac{d}{d x} \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\ln{\left(x - 3 \right)} \sin{\left(x - 3 \right)}} \]rewriteRewrite the base-exponent form using the exponential function.≈ Checked numerically
- \[ = e^{\ln{\left(x - 3 \right)} \sin{\left(x - 3 \right)}} \frac{d}{d x} \ln{\left(x - 3 \right)} \sin{\left(x - 3 \right)} \]chainApply the chain rule.✓ Proved
- \[ = \left(\ln{\left(x - 3 \right)} \frac{d}{d x} \sin{\left(x - 3 \right)} + \sin{\left(x - 3 \right)} \frac{d}{d x} \ln{\left(x - 3 \right)}\right) e^{\ln{\left(x - 3 \right)} \sin{\left(x - 3 \right)}} \]productApply the product rule to the inner expression.✓ Proved
- \[ = \left(\ln{\left(x - 3 \right)} \cos{\left(x - 3 \right)} + \frac{\sin{\left(x - 3 \right)}}{x - 3}\right) e^{\ln{\left(x - 3 \right)} \sin{\left(x - 3 \right)}} \]derivativeDifferentiate the trigonometric and logarithmic terms.✓ Proved
- \[ = \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \left(\ln{\left(x - 3 \right)} \cos{\left(x - 3 \right)} + \frac{\sin{\left(x - 3 \right)}}{x - 3}\right) \]simplifySubstitute the original function back and simplify.≈ Checked numerically
Answer \( \left(x - 3\right)^{\sin{\left(x - 3 \right)}} \left(\log{\left(x - 3 \right)} \cos{\left(x - 3 \right)} + \frac{\sin{\left(x - 3 \right)}}{x - 3}\right) \)
✓ Nihil obstat Lines: 5 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 ((x - 3)*log(x - 3)*cos(x - 3) + sin(x - 3))*((x - 3)**sin(x - 3) - exp(log(x - 3)*sin(x - 3)))/(x - 3); 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 x - 3 = 0 |
| 6 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left ((x - 3)*log(x - 3)*cos(x - 3) + sin(x - 3))*(-(x - 3)**sin(x - 3) + exp(log(x - 3)*sin(x - 3)))/(x - 3); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where x - 3 = 0 |
| 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 undefined where x - 3 = 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 (13)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-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 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18
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.