Derivative of \( \displaystyle - \frac{3 \sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \)
Problem 2.1061 · hard
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{x - 3} \frac{d}{d x} \cos{\left(x - 3 + \frac{\pi}{4} \right)} + \cos{\left(x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 3}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)} + e^{x - 3} \frac{d}{d x} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- e^{x - 3} \sin{\left(x - 3 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 3 + \frac{\pi}{4}\right) + e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]trigDifferentiate the cosine term using the chain rule.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- e^{x - 3} \sin{\left(x - 3 + \frac{\pi}{4} \right)} + e^{x - 3} \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right)}{2} \]derivative derivativeDifferentiate the inner function of the sine term. The derivative of the inner linear function is 1.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(x - 3 + \frac{\pi}{4} \right)} + \cos{\left(x - 3 + \frac{\pi}{4} \right)}\right) e^{x - 3}}{2} \]algebraFactor out the common exponential term.✓ Proved
- \[ = 3 e^{x - 3} \sin{\left(x - 3 \right)} \]simplifyUse the trigonometric identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4) or similar simplification.✓ Proved
Answer \( 3 e^{x - 3} \sin{\left(x - 3 \right)} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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) — The final simplified derivative is incorrectly stated as 3 exp(x‑3) sin(x‑3). The correct result from the steps is –3 exp(x‑3) cos(x‑3+π/4), which does not equal the given answer.qwen3.6:27b-mlx: fail (error) — Step 9 applies a trigonometric identity that is mathematically incorrect; cos(A) - sin(A) equals sqrt(2)cos(A + pi/4), not sqrt(2)cos(A + pi/2). Additionally, Step 5 is labeled 'trig' but performs differentiation (chain rule), which is a mislabeling defect.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 9 applies a trigonometric identity that is mathematically incorrect; cos(A) - sin(A) equals sqrt(2)cos(A + pi/4), not sqrt(2)cos(A + pi/2). Additionally, Step 5 is labeled 'trig' but performs differentiation (chain rule), which is a mislabeling defect.gpt-oss:20b: fail (error) 2026-09-28 — The final simplified derivative is incorrectly stated as 3 exp(x‑3) sin(x‑3). The correct result from the steps is –3 exp(x‑3) cos(x‑3+π/4), which does not equal the given answer.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the constant multiple, product, and chain rules. The final simplification step is algebraically valid and leads to the correct stated answer.gpt-oss:20b: fail (error) 2026-09-28 — Step 9 applies two rules at once: it uses the trigonometric identity cos – sin = √2 cos(⋯+π/4) and then multiplies by the existing √2 factor. This violates the one‑rule‑per‑step rule and also skips the subsequent simplification to sin, leaving the final expression incorrect.
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 2026-09-28 with SymPy 1.14.0.