Derivative of \( \displaystyle \frac{\left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \)
Problem 2.1246 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \).
- \[ \frac{d}{d x} \frac{\left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) e^{4 x - 1}}{8} \]constant-multipleFactor out the constant 1/8.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- e^{4 x - 1} \sin{\left(4 x - 1 \right)}\right)}{8} + \frac{\frac{d}{d x} e^{4 x - 1} \cos{\left(4 x - 1 \right)}}{8} \]productApply the product rule to the sum of two terms.✓ Proved
- \[ = \frac{\left(- \sin{\left(4 x - 1 \right)} + \cos{\left(4 x - 1 \right)}\right) \frac{d}{d x} e^{4 x - 1}}{8} + \frac{e^{4 x - 1} \frac{d}{d x} \left(- \sin{\left(4 x - 1 \right)}\right)}{8} + \frac{e^{4 x - 1} \frac{d}{d x} \cos{\left(4 x - 1 \right)}}{8} \]productWait, let's use the product rule on each term separately.✓ Proved
- \[ = \frac{e^{4 x - 1} \frac{d}{d x} \left(- \sin{\left(4 x - 1 \right)}\right)}{8} + \frac{e^{4 x - 1} \frac{d}{d x} \cos{\left(4 x - 1 \right)}}{8} - \frac{\sin{\left(4 x - 1 \right)} \frac{d}{d x} e^{4 x - 1}}{8} + \frac{\cos{\left(4 x - 1 \right)} \frac{d}{d x} e^{4 x - 1}}{8} \]productApply the product rule to both parts of the sum.✓ Proved
- \[ = - e^{4 x - 1} \sin{\left(4 x - 1 \right)} \]chain algebra simplify simplifyApply the chain rule to the trigonometric and exponential functions. Expand the terms in the numerator. Combine like terms. Simplify the final expression.✓ Proved
Answer \( - e^{4 x - 1} \sin{\left(4 x - 1 \right)} \)
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: fail (error) — Step 3 incorrectly applies the product rule to a sum of terms, effectively combining two distinct operations (product rule and sum rule) in one step, which violates the one‑rule‑per‑step rule. Subsequent steps propagate this error, leading to an incorrect derivative.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly labels the application of the sum rule as 'product'. Step 4 is logically incoherent, mixing product rule applications with an unexplained rearrangement of terms that does not follow from the previous step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 incorrectly labels the application of the sum rule as 'product'. Step 4 is logically incoherent, mixing product rule applications with an unexplained rearrangement of terms that does not follow from the previous step.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 incorrectly applies the product rule to a sum of terms, effectively combining two distinct operations (product rule and sum rule) in one step, which violates the one‑rule‑per‑step rule. Subsequent steps propagate this error, leading to an incorrect derivative.qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 3 is labeled 'product' but applies the sum rule to split the derivative of a sum into two terms. Step 4 is labeled 'product' but performs an incorrect algebraic expansion that does not follow the product rule structure shown in Step 5, making the logic incoherent.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 incorrectly applies the product rule to a sum; the derivative of a sum is the sum of derivatives, not a product rule. Subsequent steps propagate this mistake, leading to an incorrect application of the product rule and an overall flawed differentiation process.
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 2026-09-29 with SymPy 1.14.0.