Derivative of \( \displaystyle - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{6} \)
Problem 2.844 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{2} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{6}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\sqrt{2} \frac{d}{d x} e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{\sqrt{2} \left(e^{3 x + 1} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} \frac{d}{d x} e^{3 x + 1}\right)}{6} \]productApply the product rule.✓ Proved
- \[ = - \frac{\sqrt{2} \left(3 e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)} + e^{3 x + 1} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right)}{6} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- 3 e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + 3 e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right)}{6} \]chain trigDifferentiate the cosine term using the chain rule. The derivative of cos(u) is -sin(u) * u'.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- e^{3 x + 1} \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + e^{3 x + 1} \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right)}{2} \]constant-multiple simplifyFactor out the 3. Simplify the constant coefficient.✓ Proved
- \[ = - \frac{\sqrt{2} \left(- \sin{\left(3 x + \frac{\pi}{4} + 1 \right)} + \cos{\left(3 x + \frac{\pi}{4} + 1 \right)}\right) e^{3 x + 1}}{2} \]algebraFactor out the exponential term.✓ Proved
Answer \( e^{3 x + 1} \sin{\left(3 x + 1 \right)} \)
✓ 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 |
| 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: passqwen3.6:27b-mlx: fail (error) — The final answer provided in the task description (exp(3*x + 1)*sin(3*x + 1)) does not match the result of the derivation steps (-sqrt(2)/2 * exp(3*x + 1) * (cos(3*x + pi/4 + 1) - sin(3*x + pi/4 + 1))). Although the steps are algebraically consistent with each other, they fail to reach the stated answer, indicating a missing simplification step or an incorrect target answer. Specifically, the expression in step 9 simplifies to exp(3*x+1)*sin(3*x+1) using trigonometric identities, but this final simplification is not shown in the steps.
Senior review claude-sonnet-5-5, 2026-10-13: fail — The math is correct and every line is equivalent, including the final line against the stated answer. Step 6 is labelled 'trig' with a note about the derivative of cos, but the cosine derivative was already applied in step 5, so step 6 only moves a sign and should be labelled 'algebra'.
qwen3.6:27b-mlx: dismiss — The stated answer is correct: cos(a)-sin(a)=√2·cos(a+π/4), so with a=3x+1+π/4 line 9 reduces to exp(3x+1)·sin(3x+1), and SymPy proved the final line equal to the stated answer; the derivation lines are valid and no extra step is required.(panel model that both passed and objected): dismiss — The objection is inconsistent and unspecified, and no defect is identified.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-13 — The math is correct and every line is equivalent, including the final line against the stated answer. Step 6 is labelled 'trig' with a note about the derivative of cos, but the cosine derivative was already applied in step 5, so step 6 only moves a sign and should be labelled 'algebra'.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final answer provided in the task description (exp(3*x + 1)*sin(3*x + 1)) does not match the result of the derivation steps (-sqrt(2)/2 * exp(3*x + 1) * (cos(3*x + pi/4 + 1) - sin(3*x + pi/4 + 1))). Although the steps are algebraically consistent with each other, they fail to reach the stated answer, indicating a missing simplification step or an incorrect target answer. Specifically, the expression in step 9 simplifies to exp(3*x+1)*sin(3*x+1) using trigonometric identities, but this final simplification is not shown in the steps.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution stops at step 9 and does not reach the stated answer. Step 9 is not algebraically equal to the stated answer `exp(3*x + 1)*sin(3*x + 1)`; the solution is incomplete and incorrect as presented.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules and algebraic simplifications. The final result is algebraically equivalent to the stated answer via trigonometric identities.gpt-oss:20b: pass 2026-09-26
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.