Derivative of \( \displaystyle - \frac{5 \sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \)
Problem 2.1239 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{5 \sqrt{2} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{5 \sqrt{2} \frac{d}{d x} e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{x + 2} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 2 \right)} + \cos{\left(x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{x + 2}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- e^{x + 2} \sin{\left(x + \frac{\pi}{4} + 2 \right)} + e^{x + 2} \cos{\left(x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]derivativeDifferentiate the individual parts.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 2 \right)} + \cos{\left(x + \frac{\pi}{4} + 2 \right)}\right) e^{x + 2}}{2} \]algebraFactor out the common term exp(x + 2).✓ Proved
- \[ = 5 e^{x + 2} \sin{\left(x + 2 \right)} \]simplifySimplify the trigonometric expression.✓ Proved
Answer \( 5 e^{x + 2} \sin{\left(x + 2 \right)} \)
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. 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 |
| 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) — Step 4 applies two derivative rules at once: it replaces both Derivative(exp(x+2),x) and Derivative(cos(...),x) in a single line, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, product rule, and chain rule (labeled as derivative for the inner functions). The final simplification step is algebraically valid and correctly labeled.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the constant multiple rule, product rule, and chain rule (labeled as derivative for the inner functions). The final simplification step is algebraically valid and correctly labeled.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 applies two derivative rules at once: it replaces both Derivative(exp(x+2),x) and Derivative(cos(...),x) in a single line, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the constant multiple, product, and derivative rules. The final simplification step, while complex, is algebraically valid and correctly labeled as 'simplify'.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29
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 2026-10-03 with SymPy 1.14.0.