Derivative of \( \displaystyle \frac{\left(3 \sin{\left(2 x + 2 \right)} - 3 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \)
Problem 2.1060 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \).
- \[ \frac{d}{d x} \frac{\left(3 \sin{\left(2 x + 2 \right)} - 3 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) \frac{d}{d x} e^{2 x + 2}}{4} + \frac{3 e^{2 x + 2} \frac{d}{d x} \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right)}{4} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{2} + \frac{3 \left(2 \sin{\left(2 x + 2 \right)} + 2 \cos{\left(2 x + 2 \right)}\right) e^{2 x + 2}}{4} \]algebraDifferentiate the trigonometric and exponential terms.✓ Proved
- \[ = 3 e^{2 x + 2} \sin{\left(2 x + 2 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Simplify the final expression.✓ Proved
Answer \( 3 e^{2 x + 2} \sin{\left(2 x + 2 \right)} \)
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 |
| 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 (style) — Step 3 applies both the sum rule (to differentiate sin − cos) and the chain rule, yet it is labeled only as "chain". Step 4 also mislabels the differentiation of the trigonometric terms as "algebra" instead of "chain".qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which is a distinct rule. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions (chain rule and basic derivatives), which is a mathematical operation, not algebraic manipulation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which is a distinct rule. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions (chain rule and basic derivatives), which is a mathematical operation, not algebraic manipulation.gpt-oss:20b: fail (style) 2026-09-28 — Step 3 applies both the sum rule (to differentiate sin − cos) and the chain rule, yet it is labeled only as "chain". Step 4 also mislabels the differentiation of the trigonometric terms as "algebra" instead of "chain".qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which is a distinct rule from the chain rule. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions (chain rule and standard derivatives), which is a differentiation step, not an algebraic manipulation.gpt-oss:20b: fail (style) 2026-09-28 — Step 4 is labeled "algebra" but actually applies the chain rule to differentiate the inner functions of sin and cos. The correct label should be "chain".
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 2026-09-28 with SymPy 1.14.0.