Derivative of \( \displaystyle \frac{\left(3 \sin{\left(4 x - 1 \right)} - 3 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \)
Problem 2.478 · hard
Differentiate \( \displaystyle f(x) = \frac{3 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
- \[ \frac{d}{d x} \frac{\left(3 \sin{\left(4 x - 1 \right)} - 3 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{3 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{3 e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]sumDifferentiate the terms in the subtraction.✓ Proved
- \[ = \frac{3 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{3 \left(\frac{d}{d x} \sin{\left(4 x - 1 \right)} - \frac{d}{d x} \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]trigDifferentiate sine and cosine.✓ Proved
- \[ = \frac{3 \left(4 \sin{\left(4 x - 1 \right)} - 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} + \frac{3 \left(4 \sin{\left(4 x - 1 \right)} + 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]exponential algebraDifferentiate the exponential function. Simplify the signs.✓ Proved
- \[ = 3 e^{4 x - 1} \sin{\left(4 x - 1 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved
Answer \( 3 e^{4 x - 1} \sin{\left(4 x - 1 \right)} \)
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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 is labeled 'sum' but performs differentiation of the terms inside the derivative operator, which is not what the 'sum' rule (linearity of differentiation) alone does; it conflates the application of the derivative operator with the sum rule. More critically, Step 4 is labeled 'trig' but applies the chain rule to differentiate sin(4x-1) and cos(4x-1), which is a defect because 'chain' is the required label for differentiating composite trigonometric functions. Step 5 is labeled 'exponential' but also applies the chain rule to exp(4x-1), missing the 'chain' label.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies three derivative rules at once (for sin, cos, and exp) and is labeled incorrectly as "exponential". This violates the one‑rule‑per‑step rule and mislabels the operation.qwen3.6:27b-mlx: pass 2026-09-20 — 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) 2026-09-20 — Step 5 applies two differentiation rules at once (trigonometric and exponential) but is labeled only as "exponential". Each step must change only one thing and use a single rule label.
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.