Derivative of \( \displaystyle - \frac{3 \sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \)
Problem 2.1372 · hard
Differentiate \( \displaystyle f(x) = - \frac{3 \sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(- \frac{3 \sqrt{2} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{3 \sqrt{2} \frac{d}{d x} e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)}}{10} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{5 x - 1} \frac{d}{d x} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} + \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{5 x - 1}\right)}{10} \]productApply the product rule.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(5 x - 1\right) + e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} \left(- 5 x + \frac{\pi}{4} + 1\right)\right)}{10} \]chainApply the chain rule to both terms.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(5 e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - 5 e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right)}{10} \]derivative algebraDifferentiate the inner functions. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(e^{5 x - 1} \sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - e^{5 x - 1} \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]constant-multiple simplifyFactor out the 5. Simplify the fraction.✓ Proved
- \[ = - \frac{3 \sqrt{2} \left(\sin{\left(- 5 x + \frac{\pi}{4} + 1 \right)} - \cos{\left(- 5 x + \frac{\pi}{4} + 1 \right)}\right) e^{5 x - 1}}{2} \]algebraFactor out the exponential term.✓ Proved
Answer \( 3 e^{5 x - 1} \sin{\left(5 x - 1 \right)} \)
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 |
| 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) — Step 4 applies the chain rule to both terms of the product rule in a single step, violating the rule that each step must change only one thing. This combines two rule applications into one, which is a defect.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. The final result is algebraically equivalent to the stated answer, as verified by trigonometric identities.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules and algebraic simplifications. The final result is algebraically equivalent to the stated answer, as verified by trigonometric identities.gpt-oss:20b: fail (error) 2026-10-03 — Step 4 applies the chain rule to both terms of the product rule in a single step, violating the rule that each step must change only one thing. This combines two rule applications into one, which is a defect.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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.