Derivative of \( \displaystyle \frac{\left(5 \sin{\left(4 x - 1 \right)} - 5 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \)
Problem 2.594 · hard Video
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
- \[ \frac{d}{d x} \frac{\left(5 \sin{\left(4 x - 1 \right)} - 5 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{5 e^{4 x - 1} \frac{d}{d x} \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right)}{8} \]sumDifferentiate the terms inside the parentheses.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x - 1 \right)} - \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{5 \left(\frac{d}{d x} \sin{\left(4 x - 1 \right)} - \frac{d}{d x} \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]chainApply the chain rule to the sine and cosine terms.✓ Proved
- \[ = \frac{5 \left(4 \sin{\left(4 x - 1 \right)} - 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} + \frac{5 \left(4 \sin{\left(4 x - 1 \right)} + 4 \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]trig algebraEvaluate the derivatives of sine and cosine. Simplify the signs.✓ Proved
- \[ = 5 e^{4 x - 1} \sin{\left(4 x - 1 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Final simplification.✓ Proved
Answer \( 5 e^{4 x - 1} \sin{\left(4 x - 1 \right)} \)
✓ 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 |
| 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, 2026-10-10: pass — The math is correct: the product rule gives 5/8*(4(cos+sin)e^u + 4(sin-cos)e^u) = 5 sin(u) e^u, which matches the stated answer, and SymPy proved every line. There were no listed panel objections and no weak lines, so no rulings or checks are needed.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — The math is correct: the product rule gives 5/8*(4(cos+sin)e^u + 4(sin-cos)e^u) = 5 sin(u) e^u, which matches the stated answer, and SymPy proved every line. There were no listed panel objections and no weak lines, so no rulings or checks are needed.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'sum' but performs the product rule expansion (differentiating the product of two functions), which is a logical error in labeling and rule application. Step 4 is labeled 'chain' but performs the differentiation of sine and cosine (derivative rule) and the chain rule simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'sum' but performs the product rule expansion (distributing the derivative operator over the product), which is a violation of the single-rule-per-step constraint and mislabeling. Step 4 is labeled 'chain' but actually evaluates the derivatives of sin and cos (which should be labeled 'trig' or 'derivative'), while the chain rule application (multiplying by 4) happens in Step 5. The labels and operations are misaligned.gpt-oss:20b: pass 2026-09-21
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.