Derivative of \( \displaystyle - \frac{\sqrt{2} e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}}{2} \)
Problem 2.1342 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}}{2}\right) \]Differentiate the function with respect to x.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x + 1} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{5 x + 1}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(5 e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)} + e^{5 x + 1} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]derivativeDifferentiate the exponential part.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 5 e^{5 x + 1} \sin{\left(5 x + \frac{\pi}{4} + 1 \right)} + 5 e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]chain algebraDifferentiate the cosine part using the chain rule. Distribute the 5.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- e^{5 x + 1} \sin{\left(5 x + \frac{\pi}{4} + 1 \right)} + e^{5 x + 1} \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]algebraFactor out the 5.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(5 x + \frac{\pi}{4} + 1 \right)}\right) e^{5 x + 1}}{2} \]algebraFactor out the exponential term.✓ Proved
- \[ = 5 e^{5 x + 1} \sin{\left(5 x + 1 \right)} \]simplify simplifyUse the trigonometric identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4). Simplify the expression by combining terms.✓ Proved
Answer \( 5 e^{5 x + 1} \sin{\left(5 x + 1 \right)} \)
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 |
| 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 |
| 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 derivative is incorrectly simplified. The correct result after applying the identity is -5*exp(5*x+1)*cos(5*x+pi/2+1), not 5*exp(5*x+1)*sin(5*x+1). The sign and trigonometric function are wrong.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single rule application as required.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and trigonometric identities to reach the stated answer. Each step isolates a single rule application as required.gpt-oss:20b: fail (error) 2026-10-03 — The final derivative is incorrectly simplified. The correct result after applying the identity is -5*exp(5*x+1)*cos(5*x+pi/2+1), not 5*exp(5*x+1)*sin(5*x+1). The sign and trigonometric function are wrong.qwen3.6:27b-mlx: inconclusive 2026-09-30 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 9 applies the identity cos(A) - sin(A) = sqrt(2)cos(A + pi/4) to the term (cos(5*x + pi/4 + 1) - sin(5*x + pi/4 + 1)). This is incorrect;gpt-oss:20b: inconclusive 2026-09-30 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative is incorrectly simplified to \u22125e^{5x+1}\cos(5x+1+\pi/2), which is not equal to the correct result 5e^{5x+1}\sin(5x+1). The sign
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.