Derivative of \( \displaystyle \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{10} \)
Problem 2.1297 · hard
Differentiate \( \displaystyle f(x) = \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{10} \).
- \[ \frac{d}{d x} \frac{\sqrt{2} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{10} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\sqrt{2} \frac{d}{d x} e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}}{10} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{5 x + 2} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} e^{5 x + 2}\right)}{10} \]productApply the product rule.✓ Proved
- \[ = \frac{\sqrt{2} \left(e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} \left(5 x + 2\right) + e^{5 x + 2} \frac{d}{d x} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{10} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} \frac{d}{d x} \left(5 x + \frac{\pi}{4} + 2\right) + 5 e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{10} \]trigApply the chain rule to the cosine term.✓ Proved
- \[ = \frac{\sqrt{2} \left(- 5 e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} + 5 e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{10} \]constant algebraDifferentiate the inner linear function. Distribute the 5.✓ Proved
- \[ = \frac{\sqrt{2} \left(- e^{5 x + 2} \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} + e^{5 x + 2} \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right)}{2} \]algebra simplifyFactor out the 5. Simplify the constant coefficient.✓ Proved
- \[ = \frac{\sqrt{2} \left(- \sin{\left(5 x + \frac{\pi}{4} + 2 \right)} + \cos{\left(5 x + \frac{\pi}{4} + 2 \right)}\right) e^{5 x + 2}}{2} \]algebraFactor out the exponential term.✓ Proved
Answer \( - e^{5 x + 2} \sin{\left(5 x + 2 \right)} \)
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 |
| 10 | ✓ 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) — Step 5 is labeled 'trig' but applies the chain rule to the cosine term (introducing Derivative(5*x + pi/4 + 2, x)). The label 'trig' is incorrect for a step that performs differentiation via the chain rule; it should be labeled 'chain'. Additionally, the note says 'Apply the chain rule', which contradicts the label 'trig', but the label itself is the defect as it does not name the rule applied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is labeled 'trig' but applies the chain rule to the cosine term (introducing Derivative(5*x + pi/4 + 2, x)). The label 'trig' is incorrect for a step that performs differentiation via the chain rule; it should be labeled 'chain'. Additionally, the note says 'Apply the chain rule', which contradicts the label 'trig', but the label itself is the defect as it does not name the rule applied.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (misleading) 2026-09-30 — Step 5 is labeled 'trig' but the note claims it applies the chain rule; the step actually applies the derivative of cosine (trig) and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, the final answer does not match the stated answer, as the trigonometric simplification to -sin(5x+2) is not performed.gpt-oss:20b: fail (error) 2026-09-30 — The final result differs from the stated answer; the solution omits the cosine term and the constant factor, so the derivative is incorrect.
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.