Derivative of \( \displaystyle - \frac{5 \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6} \)
Problem 2.71 · hard
Differentiate \( \displaystyle f(x) = - \frac{5 \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6} \).
- \[ \frac{d}{d x} \left(- \frac{5 \sqrt{2} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{5 \sqrt{2} \frac{d}{d x} e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}}{6} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(e^{3 x} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)} \frac{d}{d x} e^{3 x}\right)}{6} \]productApply the product rule to the remaining terms.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(3 e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)} + e^{3 x} \frac{d}{d x} \cos{\left(3 x + \frac{\pi}{4} \right)}\right)}{6} \]derivativeDifferentiate the exponential term.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- 3 e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)} + 3 e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}\right)}{6} \]chain algebraDifferentiate the cosine term using the chain rule. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)} + e^{3 x} \cos{\left(3 x + \frac{\pi}{4} \right)}\right)}{2} \]algebra simplifyFactor out the common factor of 3. Simplify the coefficient fraction.✓ Proved
- \[ = - \frac{5 \sqrt{2} \left(- \sin{\left(3 x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}\right) e^{3 x}}{2} \]algebraFactor out the exponential term.✓ Proved
Answer \( 5 e^{3 x} \sin{\left(3 x \right)} \)
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: passdeepseek-r1:70b: fail (error) — The final answer does not match the derived expression, indicating an error in the differentiation process.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. The final result matches the stated answer.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. The final result matches the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. The final step implicitly uses trigonometric identities to reach the stated answer, which is consistent with the problem statement.gpt-oss:20b: fail (error) 2026-09-20 — The final answer 5*exp(3*x)*sin(3*x) does not match the derivative obtained in the steps. The correct derivative is -5*sqrt(2)/2*exp(3*x)*(cos(3*x+pi/4)-sin(3*x+pi/4)).qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple, product, chain, and derivative rules in a step-by-step manner. The final simplification to the stated answer is mathematically sound, though the last step (trigonometric identity simplification) is not explicitly shown as a separate step, the previous steps are valid.deepseek-r1:70b: fail (error) 2026-09-19 — The final answer does not match the derived expression, indicating an error in the differentiation process.gpt-oss:20b: fail (error) 2026-09-19 — The final derivative is incorrect: the √2 factor should cancel when using the identity cos(3x+π/4)=(cos3x−sin3x)/√2, yielding 5 exp(3x) sin(3x). The solution stops short of this simplification, leaving an extra √2 and an incorrect sign.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. The final result matches the stated answer, as the trigonometric simplification of the derivative expression yields 5*exp(3*x)*sin(3*x).deepseek-r1:70b: fail (style) 2026-09-19 — The first step lacks a rule label, which could confuse students. Additionally, the final simplified expression does not match the stated answer, potentially misleading students.gpt-oss:20b: fail (error) 2026-09-19 — The final derivative does not match the correct result; the algebraic simplification is incorrect and the answer is not equivalent to the true derivative of the given function.gpt-oss:20b: fail 2026-09-17 — The steps stop at a factored form and do not simplify to the stated answer 5*exp(3*x)*sin(3*x).deepseek-r1:70b: pass 2026-09-17
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.