Derivative of \( \displaystyle - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \)
Problem 2.207 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{5 \sqrt{2} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{5 \sqrt{2} \frac{d}{d x} e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{x + 1} \frac{d}{d x} \cos{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{x + 1}\right)}{2} \]productApply the product rule to the remaining terms.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- e^{x + 1} \sin{\left(x + \frac{\pi}{4} + 1 \right)} + e^{x + 1} \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]derivativeDifferentiate the exponential and cosine terms.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- \sin{\left(x + \frac{\pi}{4} + 1 \right)} + \cos{\left(x + \frac{\pi}{4} + 1 \right)}\right) e^{x + 1}}{2} \]algebraFactor out the common exponential term.✓ Proved
- \[ = 5 e^{x + 1} \sin{\left(x + 1 \right)} \]simplify simplifyUse the trigonometric identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4). Simplify the expression by combining the sqrt(2) terms.✓ Proved
Answer \( 5 e^{x + 1} \sin{\left(x + 1 \right)} \)
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 |
| 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative computed in the solution is \u22125 e^{x+1}\cos(x+π/2+1), which is algebraically correct, but it does not match the stated answer 5 edeepseek-r1:70b: fail (error) — The first step lacks a rule label, which is required by the contract.qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: inconclusive 2026-09-20 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative computed in the solution is \u22125 e^{x+1}\cos(x+π/2+1), which is algebraically correct, but it does not match the stated answer 5 eqwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid algebraic and trigonometric simplifications to reach the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple, product, and derivative rules, followed by valid trigonometric simplifications to reach the stated answer.deepseek-r1:70b: fail (error) 2026-09-19 — The first step lacks a rule label, which is required by the contract.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the constant-multiple, product, and derivative rules, followed by valid algebraic and trigonometric simplifications to reach the stated answer.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 4 is mislabeled as a derivative step—it merely repeats the previous expression without applying any rule, which could mislead a student about what was actually done.deepseek-r1:70b: fail 2026-09-17 — The final answer has an incorrect positive sign; the correct derivative should be negative.
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.