Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{3} \)
Problem 2.964 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{3}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{3 x}\right)}{3} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(3 e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)} + e^{3 x} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} \right)}\right)}{3} \]derivativeDifferentiate the exponential part.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)} + 3 e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}\right)}{3} \]chain algebraDifferentiate the cosine part using the chain rule. Simplify the expression inside the parentheses.✓ Proved
- \[ = \sqrt{2} e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)} - \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)} \]algebraDistribute the constant factor.✓ Proved
- \[ = \sqrt{2} \left(\sin{\left(3 x + \frac{\pi}{4} \right)} - \cos{\left(3 x + \frac{\pi}{4} \right)}\right) e^{3 x} \]simplifyFactor out the common terms.✓ Proved
Answer \( 2 e^{3 x} \sin{\left(3 x \right)} \)
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 |
| 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 simplification is missing a step: \(\sqrt{2}\,e^{3x}\bigl(\sin(3x+\pi/4)-\cos(3x+\pi/4)\bigr)\) simplifies to \(2e^{3x}\sin(3x)\). The solution stops at the unsimplified expression, so the stated answer is not justified.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. The final result matches the stated answer via trigonometric identities.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications. The final result matches the stated answer via trigonometric identities.gpt-oss:20b: fail (error) 2026-09-27 — The final simplification is missing a step: \(\sqrt{2}\,e^{3x}\bigl(\sin(3x+\pi/4)-\cos(3x+\pi/4)\bigr)\) simplifies to \(2e^{3x}\sin(3x)\). The solution stops at the unsimplified expression, so the stated answer is not justified.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final answer provided (2*exp(3*x)*sin(3*x)) is mathematically incorrect; the correct derivative is 2*exp(3*x)*sin(3*x + pi/2) or 2*exp(3*x)*cos(3*x). The steps correctly derive an intermediate form, but the solution fails to simplify that form to the stated answer, and the stated answer itself is wrong.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.