Derivative of \( \displaystyle \frac{\left(3 \sin{\left(2 x - 3 \right)} - 3 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \)
Problem 2.1312 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \).
- \[ \frac{d}{d x} \frac{\left(3 \sin{\left(2 x - 3 \right)} - 3 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{4} + \frac{3 e^{2 x - 3} \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right)}{4} \]chainApply the chain rule to each part.✓ Proved
- \[ = \frac{3 \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} + \frac{3 \left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{4} \]algebraDifferentiate the trigonometric and exponential terms.✓ Proved
- \[ = 3 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \]algebra simplify simplifyDistribute the terms. Combine like terms. Simplify the final expression.✓ Proved
Answer \( 3 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \)
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: passqwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions (trigonometric and exponential derivatives), which should be labeled 'derivative' or 'chain'. The labels are swapped and misapplied.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions (trigonometric and exponential derivatives), which should be labeled 'derivative' or 'chain'. The labels are swapped and misapplied.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-30 — Step 3 is labeled 'chain' but applies the product rule expansion (from step 2) and computes derivatives; the label should be 'product' or 'derivative'. Step 4 is labeled 'algebra' but performs differentiation (chain rule on inner terms), so the label should be 'chain' or 'derivative'.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.