Derivative of \( \displaystyle \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \)
Problem 2.656 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]constant-multiplePull out the constant factor 1/2.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{e^{2 x - 3} \frac{d}{d x} \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right)}{2} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{\left(\frac{d}{d x} \sin{\left(2 x - 3 \right)} - \frac{d}{d x} \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]sumApply the sum rule to the first term.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) \frac{d}{d x} e^{2 x - 3}}{2} + \frac{\left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]trig algebraDifferentiate the sine and cosine terms using the chain rule. Distribute the 2 into the parentheses.✓ Proved
- \[ = \left(\sin{\left(2 x - 3 \right)} - \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3} + \frac{\left(2 \sin{\left(2 x - 3 \right)} + 2 \cos{\left(2 x - 3 \right)}\right) e^{2 x - 3}}{2} \]exponentialDifferentiate the exponential term.✓ Proved
- \[ = 2 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \]algebra algebra simplify simplifyFactor out the common term exp(2*x - 3). Distribute the 2. Combine like terms. Simplify the final expression.✓ Proved
Answer \( 2 e^{2 x - 3} \sin{\left(2 x - 3 \right)} \)
✓ 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. How we verify
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 |
| 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: pass
Senior review claude-sonnet-5-5, 2026-10-11: pass — With u = 2x-3, f' = 1/2*[(2cos u + 2sin u)e^u + (sin u - cos u)*2e^u] = 2 sin(u) e^u, which matches the stated answer. Every line was proved equal to the previous one, and the rule labels are all legitimate. The panel listed no specific objection to rule on.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — With u = 2x-3, f' = 1/2*[(2cos u + 2sin u)e^u + (sin u - cos u)*2e^u] = 2 sin(u) e^u, which matches the stated answer. Every line was proved equal to the previous one, and the rule labels are all legitimate. The panel listed no specific objection to rule on.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 applies the chain rule to differentiate sine and cosine but is labeled 'trig', which is not the correct label for differentiation. Additionally, the note claims the chain rule is used, but the label 'trig' does not reflect this operation, violating the contract that the label must name the rule applied.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 applies the chain rule to differentiate sin(2x-3) and cos(2x-3) but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities, not differentiation rules; the correct label should be 'chain' (or 'derivative' if chain is not distinguished, but 'chain' is in the vocabulary). Additionally, the note claims to use the chain rule, but the label does not match the action, violating the contract that the label must name the rule applied.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies both the chain rule and the trig derivative rule, but it is labeled only as "trig". Each step must apply exactly one rule, so this is a defect.
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.