Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{2} \)
Problem 2.1310 · hard
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x - 3} \frac{d}{d x} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} + \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} e^{3 x - 3}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(3 e^{3 x - 3} \sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} - 3 e^{3 x - 3} \cos{\left(- 3 x + \frac{\pi}{4} + 3 \right)}\right)}{2} \]chain algebraApply the chain rule to both terms. Simplify the expression by distributing the negative sign.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(\sin{\left(- 3 x + \frac{\pi}{4} + 3 \right)} - \cos{\left(- 3 x + \frac{\pi}{4} + 3 \right)}\right) e^{3 x - 3}}{2} \]algebraFactor out the common term 3*exp(3*x - 3).✓ Proved
Answer \( 3 e^{3 x - 3} \sin{\left(3 x - 3 \right)} \)
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: fail (error) — The final answer 3*exp(3*x - 3)*sin(3*x - 3) does not match the derivative computed in the steps. The correct derivative is -3*sqrt(2)/2*exp(3*x - 3)*(sin(-3*x + pi/4 + 3) - cos(-3*x + pi/4 + 3)).qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Although the final expression differs in form from the stated answer, they are mathematically equivalent (via trigonometric identities), and the steps provided are logically sound and correctly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. Although the final expression differs in form from the stated answer, they are mathematically equivalent (via trigonometric identities), and the steps provided are logically sound and correctly labeled.gpt-oss:20b: fail (error) 2026-10-03 — The final answer 3*exp(3*x - 3)*sin(3*x - 3) does not match the derivative computed in the steps. The correct derivative is -3*sqrt(2)/2*exp(3*x - 3)*(sin(-3*x + pi/4 + 3) - cos(-3*x + pi/4 + 3)).qwen3.6:27b-mlx: fail (error) 2026-09-30 — The solution stops at step 6, which is an intermediate expression, rather than providing the stated final answer. The transition from step 6 to the stated answer requires trigonometric simplification (using sum/difference identities) which is not shown as a step.gpt-oss:20b: fail (error) 2026-09-30 — The final derivative expression does not match the stated answer and is incorrect. The steps produce -3*sqrt(2)/2*exp(3*x-3)*(sin(-3*x+pi/4+3)-cos(-3*x+pi/4+3)), which simplifies to a different function than 3*exp(3*x-3)*sin(3*x-3).
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.