Derivative of \( \displaystyle \frac{\left(5 \sin{\left(4 x - 3 \right)} - 5 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \)
Problem 2.1327 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \).
- \[ \frac{d}{d x} \frac{\left(5 \sin{\left(4 x - 3 \right)} - 5 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{8} + \frac{5 e^{4 x - 3} \frac{d}{d x} \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right)}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) \frac{d}{d x} e^{4 x - 3}}{8} + \frac{5 \left(\frac{d}{d x} \sin{\left(4 x - 3 \right)} - \frac{d}{d x} \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x - 3 \right)} - \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{2} + \frac{5 \left(4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]chainApply the chain rule to each term.✓ Proved
- \[ = \frac{5 \left(4 \sin{\left(4 x - 3 \right)} - 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} + \frac{5 \left(4 \sin{\left(4 x - 3 \right)} + 4 \cos{\left(4 x - 3 \right)}\right) e^{4 x - 3}}{8} \]algebraSimplify the signs.✓ Proved
- \[ = 5 e^{4 x - 3} \sin{\left(4 x - 3 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved
Answer \( 5 e^{4 x - 3} \sin{\left(4 x - 3 \right)} \)
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 |
| 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 three different functions (sin, cos, and exp) in one line, violating the rule that each step must change only one thing. This multi‑rule application is a defect.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 (4)
qwen3.6:27b-mlx: pass 2026-10-03 — 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) 2026-10-03 — Step 5 applies the chain rule to three different functions (sin, cos, and exp) in one line, violating the rule that each step must change only one thing. This multi‑rule application is a defect.qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.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.