Derivative of \( \displaystyle \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \)
Problem 2.860 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} \frac{e^{2 x + 2}}{4} + \frac{e^{2 x + 2} \frac{d}{d x} \left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} + \frac{e^{2 x + 2} \frac{d}{d x} \left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right)}{4} \]derivative algebraDifferentiate the second part of the product. Simplify the fraction.✓ Proved
- \[ = \frac{\left(- 2 \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} + \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \]derivative algebraDifferentiate the sum of trigonometric functions using the chain rule. Factor out -2 from the first term.✓ Proved
- \[ = \frac{\left(- \sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} + \frac{\left(- \sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \]algebraSimplify the fraction 2/4 to 1/2.✓ Proved
- \[ = - e^{2 x + 2} \sin{\left(2 x + 2 \right)} \]algebra simplify simplifyFactor out the common term (exp(2*x + 2) / 2). Combine like terms inside the parentheses. Final simplification.✓ Proved
Answer \( - e^{2 x + 2} \sin{\left(2 x + 2 \right)} \)
✓ 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.
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 (style) — Step 5 applies both the sum rule and the chain rule to differentiate the trigonometric terms, violating the one-rule-per-step constraint. Step 3 also implicitly applies the constant-multiple rule alongside the derivative of the exponential, which should be separated.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 applies both the sum rule and the chain rule to differentiate the trigonometric terms, violating the one-rule-per-step constraint. Step 3 also implicitly applies the constant-multiple rule alongside the derivative of the exponential, which should be separated.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 applies the chain rule to differentiate the trigonometric terms, but is labeled 'derivative'. While 'derivative' is acceptable for basic forms, the explicit use of the chain rule (differentiating the inner function 2*x+2) makes the label 'chain' more precise and required by the contract's distinction between basic derivatives and composite functions. Additionally, Step 3 differentiates exp(2*x+2)/4; while the result is correct, the step combines the constant multiple rule and the chain rule/exp derivative into one step labeled 'derivative', which is slightly ambiguous but acceptable under 'derivative' if viewed as a known form. However, Step 5 is the clearer violation: it performs a chain rule differentiation (derivative of sin(u) is cos(u)*u') but labels it 'derivative'. The contract lists 'chain' as a distinct label. Using 'derivative' for a step that explicitly relies on the chain rule (inner derivative 2) is a labeling defect.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.