Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \)
Problem 2.298 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} \frac{e^{2 x + 2}}{2} + \frac{e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right)}{2} + \frac{e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{2} \]chainApply the chain rule to the exponential term.✓ Proved
- \[ = \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} + \frac{e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{2} \]derivative algebraDifferentiate the inner function 2*x + 2. Simplify the constant multiplication.✓ Proved
- \[ = \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} + \frac{\left(\frac{d}{d x} \sin{\left(2 x + 2 \right)} - \frac{d}{d x} \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} + \frac{\left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Simplify the signs.✓ Proved
- \[ = \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} + \left(\sin{\left(2 x + 2 \right)} + \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2} \]algebraDistribute the 1/2 into the parentheses.✓ Proved
- \[ = 2 e^{2 x + 2} \sin{\left(2 x + 2 \right)} \]algebra simplify algebraFactor out the common exponential term. Combine like terms inside the parentheses. Final simplification.✓ Proved
Answer \( 2 e^{2 x + 2} \sin{\left(2 x + 2 \right)} \)
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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: 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 (12)
qwen3.6:27b-mlx: pass 2026-09-20 — 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 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 7 applies the chain rule to differentiate the trigonometric terms but is labeled 'trig'. While 'trig' is in the vocabulary, the step explicitly performs differentiation (unfolding d/dx), which should be labeled 'derivative' or 'chain' to accurately reflect the operation performed. Labeling a differentiation step as merely 'trig' is misleading regarding the rule applied.gpt-oss:20b: fail (error) 2026-09-20 — Step 7 applies both the chain rule and the trig derivative rule in a single step, violating the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the product rule, chain rule, and trigonometric derivatives in separate steps. The algebraic simplifications are valid and clearly labeled.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the product rule, chain rule, and trigonometric derivatives in separate steps. The algebraic simplifications are valid and clearly labeled.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-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.