Derivative of \( \displaystyle - \frac{3 \sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{10} \)
Problem 2.1229 · hard
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{10} \).
- \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{10}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{10} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{5 x - 3} \frac{d}{d x} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} + \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} e^{5 x - 3}\right)}{10} \]productApply the product rule.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(5 e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} + e^{5 x - 3} \frac{d}{d x} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}\right)}{10} \]derivativeDifferentiate the exponential part.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(5 e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} - 5 e^{5 x - 3} \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)}\right)}{10} \]chain algebraDifferentiate the sine part using the chain rule. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} - e^{5 x - 3} \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)}\right)}{2} \]constant-multiple simplifyDistribute the constant factor. Simplify the fraction.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(\sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} - \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)}\right) e^{5 x - 3}}{2} \]algebraFactor out the common exponential term.✓ Proved
Answer \( 3 e^{5 x - 3} \sin{\left(5 x - 3 \right)} \)
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 |
| 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 |
| 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 given in the problem statement (3*exp(5*x - 3)*sin(5*x - 3)) does not match the derivative computed in the steps. The steps correctly produce -3*sqrt(2)/2 * exp(5*x - 3)*(sin(-5*x + pi/4 + 3) - cos(-5*x + pi/4 + 3)), which is not equivalent to the stated answer. This is a mathematical error in the solution.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and algebraic simplifications. The final expression is algebraically equivalent to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the product rule, chain rule, and algebraic simplifications. The final expression is algebraically equivalent to the stated answer.gpt-oss:20b: fail (error) 2026-09-29 — The final answer given in the problem statement (3*exp(5*x - 3)*sin(5*x - 3)) does not match the derivative computed in the steps. The steps correctly produce -3*sqrt(2)/2 * exp(5*x - 3)*(sin(-5*x + pi/4 + 3) - cos(-5*x + pi/4 + 3)), which is not equivalent to the stated answer. This is a mathematical error in the solution.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The final answer provided in the 'Stated answer' field is mathematically incorrect (it is missing the cosine term and has the wrong coefficient), making the solution fail to solve the task. Additionally, Step 7 applies distribution (algebra) but is labeled 'constant-multiple', which is a labeling defect.gpt-oss:20b: pass 2026-09-29
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.