Derivative of \( \displaystyle - \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \)
Problem 2.653 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}}{2} \]constant-multiplePull out the constant coefficient.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} e^{x - 1}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \frac{d}{d x} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} \frac{d}{d x} \left(x - 1 + \frac{\pi}{4}\right) + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]trigDifferentiate the cosine term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{x - 1} \sin{\left(x - 1 + \frac{\pi}{4} \right)} + e^{x - 1} \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right)}{2} \]derivative algebraDifferentiate the inner function of the chain rule. Distribute the negative sign.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- \sin{\left(x - 1 + \frac{\pi}{4} \right)} + \cos{\left(x - 1 + \frac{\pi}{4} \right)}\right) e^{x - 1}}{2} \]algebraFactor out the exponential term.✓ Proved
- \[ = e^{x - 1} \sin{\left(x - 1 \right)} \]algebra algebra simplify simplifyExpand the trigonometric terms using angle addition formulas. Substitute the value of cos(pi/4) and sin(pi/4). Combine like terms inside the parentheses. Simplify the final expression.✓ Proved
Answer \( e^{x - 1} \sin{\left(x - 1 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and trigonometric identities. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-11: pass — Every line is correct and SymPy proved each step. The derivative reduces to exp(x-1)*sin(x-1), which matches the stated answer. The rule labels are legitimate, so nothing needs fixing. The panel listed no specific objection, so there is nothing to rule on.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — Every line is correct and SymPy proved each step. The derivative reduces to exp(x-1)*sin(x-1), which matches the stated answer. The rule labels are legitimate, so nothing needs fixing. The panel listed no specific objection, so there is nothing to rule on.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules and trigonometric identities. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: inconclusive 2026-09-21 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 9 incorrectly expands cos(A+B) and sin(A+B). The expansion of cos(x-1+pi/4) should be cos(x-1)cos(pi/4) - sin(x-1)sin(pi/4), but the step writegpt-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.