Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{2} \)
Problem 2.1242 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{2}\right) \]Differentiate the function with respect to x.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x + 2} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{3 x + 2}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x + 2} \sin{\left(3 x + \frac{\pi}{4} + 2 \right)} + 3 e^{3 x + 2} \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]chainApply the chain rule to both terms.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(- \sin{\left(3 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 2 \right)}\right) e^{3 x + 2}}{2} \]algebra simplifyFactor out common terms. Simplify the expression.✓ Proved
Answer \( 3 e^{3 x + 2} \sin{\left(3 x + 2 \right)} \)
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.
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final result does not match the derivative computed in the preceding steps. The correct derivative is \(-\frac{3\sqrt{2}}{2}\exp(3x+2)(\cos(3x+\pi/4+2qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplification. The final result is mathematically equivalent to the stated answer via trigonometric identities.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplification. The final result is mathematically equivalent to the stated answer via trigonometric identities.gpt-oss:20b: inconclusive 2026-09-29 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final result does not match the derivative computed in the preceding steps. The correct derivative is \(-\frac{3\sqrt{2}}{2}\exp(3x+2)(\cos(3x+\pi/4+2qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules step-by-step. The final result is algebraically equivalent to the stated answer via trigonometric identities, and no steps violate the one-rule-per-step constraint.gpt-oss:20b: inconclusive 2026-09-29 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final derivative expression does not match the stated answer. The correct derivative of \(-\sqrt{2}\,e^{3x+2}\cos(3x+\pi/4+2)/2\) is \(-\tfrac{3\sqrt{
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 2026-09-29 with SymPy 1.14.0.