Derivative of \( \displaystyle \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \)
Problem 2.861 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]Start with the derivative of the function.✓ Proved
- \[ = \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) \frac{d}{d x} \frac{e^{4 x + 2}}{8} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2} \frac{d}{d x} \left(4 x + 2\right)}{8} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]constant-multiplePull out the constant 1/8 from the second term's derivative.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} + \frac{e^{4 x + 2} \frac{d}{d x} \left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)}{8} \]derivativeDifferentiate the inner function 4*x + 2.✓ Proved
- \[ = \frac{\left(- 4 \sin{\left(4 x + 2 \right)} - 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} + \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} \]trigDifferentiate the sine and cosine terms using the chain rule.✓ Proved
- \[ = \frac{\left(- 4 \sin{\left(4 x + 2 \right)} - 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} + \frac{\left(- 4 \sin{\left(4 x + 2 \right)} + 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]algebraFactor out -4 from the first term.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} + \frac{\left(- \sin{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} \]simplifySimplify the fractions by dividing 4/8.✓ Proved
- \[ = - e^{4 x + 2} \sin{\left(4 x + 2 \right)} \]algebra simplify simplifyFactor out the common term exp(4*x + 2)/2. Combine like terms inside the parentheses. Multiply the remaining terms to get the final result.✓ Proved
Answer \( - e^{4 x + 2} \sin{\left(4 x + 2 \right)} \)
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: pass — The solution correctly applies the product rule, chain rule, and trigonometric derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and trigonometric derivatives in a step-by-step manner. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 5 applies the chain rule to differentiate the sine and cosine terms but labels the step 'trig'. The label 'trig' is not in the allowed vocabulary for differentiation rules (which includes 'chain'), and the step performs differentiation, not just trigonometric simplification.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 2026-09-26 with SymPy 1.14.0.