Derivative of \( \displaystyle - \frac{\sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5} \)
Problem 2.185 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5}\right) \]constantStart with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x - 3} \frac{d}{d x} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} + \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} e^{5 x - 3}\right)}{5} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} \left(5 x - 3\right) + e^{5 x - 3} \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)} \frac{d}{d x} \left(- 5 x + \frac{\pi}{4} + 3\right)\right)}{5} \]chainApply the chain rule to both terms.✓ Proved
- \[ = - \frac{\sqrt{2} \left(5 e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} - 5 e^{5 x - 3} \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)}\right)}{5} \]derivative algebraEvaluate the derivatives of the inner functions. Distribute the derivative results.✓ Proved
- \[ = - \sqrt{2} e^{5 x - 3} \sin{\left(- 5 x + \frac{\pi}{4} + 3 \right)} + \sqrt{2} e^{5 x - 3} \cos{\left(- 5 x + \frac{\pi}{4} + 3 \right)} \]simplifySimplify the expression by canceling the 5.✓ Proved
Answer \( 2 e^{5 x - 3} \sin{\left(5 x - 3 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify
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 |
| 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: passdeepseek-r1:70b: fail (misleading) — The simplification step does not show how sine and cosine terms combine into the final sine function, which is misleading for understanding.qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-05: pass — Every line was proved equivalent by SymPy. The derivative is correct: with t=5x-3, sin(π/4-t)-cos(π/4-t) = -√2 sin t, so the result is 2e^t sin t, which matches the stated answer. The rule labels are all legitimate. No specific panel objection was listed, so there is nothing to rule on.
Every verdict on record (16)
claude-sonnet-5-5: pass 2026-10-05 — Every line was proved equivalent by SymPy. The derivative is correct: with t=5x-3, sin(π/4-t)-cos(π/4-t) = -√2 sin t, so the result is 2e^t sin t, which matches the stated answer. The rule labels are all legitimate. No specific panel objection was listed, so there is nothing to rule on.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The final simplification to the stated answer is algebraically valid (using trigonometric identities), though the final step is omitted from the list, the provided steps are correct.gpt-oss:20b: fail (style) 2026-09-20 — Step 4 applies the chain rule twice in one line (to both the exponential and the sine term), violating the rule that each step must change only one thing.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: fail (misleading) 2026-09-19 — The simplification step does not show how sine and cosine terms combine into the final sine function, which is misleading for understanding.gpt-oss:20b: fail (error) 2026-09-19 — The final stated answer 2*exp(5*x - 3)*sin(5*x - 3) does not match the derivative computed in the steps, which simplifies to -sqrt(2)*exp(5*x - 3)*sin(-5*x + pi/4 + 3)+sqrt(2)*exp(5*x - 3)*cos(-5*x + pi/4 + 3).qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 4 applies the chain rule to both terms of the sum simultaneously, violating the 'one change per step' constraint. It should be split into two steps or labeled as applying the rule to the whole expression, but the label 'chain' implies a single structural transformation, whereas two distinct inner derivatives are computed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — The derivative computed in the steps is -√2 exp(5x‑3) sin(−5x+π/4+3)+√2 exp(5x‑3) cos(−5x+π/4+3), which is not algebraically equivalent to the stated answer 2 exp(5x‑3) sin(5x‑3). The solution therefore contains a mathematical error.qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: fail (misleading) 2026-09-18 — The simplification step incorrectly handles the negative arguments in sine and cosine, leading to an erroneous result.gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — The final answer 2*exp(5*x - 3)*sin(5*x - 3) does not match the derivative obtained in the steps; the correct derivative is sqrt(2)*exp(5*x - 3)*(sin(5*x - pi/4 - 3)+cos(5*x - pi/4 - 3)).deepseek-r1:70b: fail 2026-09-17 — The final simplified expression does not match the stated answer, indicating an error in the simplification process.
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 2026-09-26 with SymPy 1.14.0.