Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \)
Problem 2.397 · 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
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]constant-multiplePull out the constant factor 1/4.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} e^{2 x + 2}}{4} + \frac{e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} e^{2 x + 2}}{4} + \frac{\left(\frac{d}{d x} \sin{\left(2 x + 2 \right)} - \frac{d}{d x} \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} + \frac{\left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]chainApply the chain rule to the trigonometric and exponential terms.✓ Proved
- \[ = \frac{\left(2 \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} + \frac{\left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]algebraSimplify the signs inside the parentheses.✓ Proved
- \[ = e^{2 x + 2} \sin{\left(2 x + 2 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Simplify the final expression.✓ Proved
Answer \( e^{2 x + 2} \sin{\left(2 x + 2 \right)} \)
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 |
| 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: fail (error) — Step 5 applies the chain rule to multiple terms at once (to both the sine and cosine inside the first product and to the exponential in the second product). Each application of the chain rule must be a separate step; combining them violates the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately describe the operations performed, and the final result is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately describe the operations performed, and the final result is correct.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the chain rule to multiple terms at once (to both the sine and cosine inside the first product and to the exponential in the second product). Each application of the chain rule must be a separate step; combining them violates the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. The labels accurately reflect the operations performed, and the final result is correct.gpt-oss:20b: fail (style) 2026-09-20 — Step 5 applies the chain rule to several terms (sin, cos, and exp) in a single line, violating the rule that each step must change only one thing. Separate steps for each derivative would be required.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20
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.