Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \)
Problem 2.1249 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \]constant-multipleFactor out the constant 1/4.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{\left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) \frac{d}{d x} e^{4 x + 1}}{4} + \frac{e^{4 x + 1} \frac{d}{d x} \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right)}{4} \]chainApply the chain rule to each part.✓ Proved
- \[ = \left(\sin{\left(4 x + 1 \right)} - \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1} + \frac{\left(4 \sin{\left(4 x + 1 \right)} + 4 \cos{\left(4 x + 1 \right)}\right) e^{4 x + 1}}{4} \]algebraDifferentiate the sine and cosine terms using the chain rule.✓ Proved
- \[ = 2 e^{4 x + 1} \sin{\left(4 x + 1 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Simplify the final expression.✓ Proved
Answer \( 2 e^{4 x + 1} \sin{\left(4 x + 1 \right)} \)
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. A reviewer's objection to the explanation stands; see 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 |
| 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 labels the application of the product rule as "chain"; Step 4 incorrectly labels the differentiation of the sine and cosine terms as "algebra" when it is a chain rule application. These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the product rule expansion (Step 2's label was 'product' but the step didn't expand). Step 4 is labeled 'algebra' but performs differentiation (chain rule on sin/cos/exp). The labels are swapped and incorrect for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 is labeled 'chain' but performs the product rule expansion (Step 2's label was 'product' but the step didn't expand). Step 4 is labeled 'algebra' but performs differentiation (chain rule on sin/cos/exp). The labels are swapped and incorrect for the operations performed.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 incorrectly labels the application of the product rule as "chain"; Step 4 incorrectly labels the differentiation of the sine and cosine terms as "algebra" when it is a chain rule application. These mislabelings violate the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 is labeled 'chain' but performs the product rule expansion (Step 2's label) and leaves the derivatives unevaluated. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions using the chain rule. The labels for steps 3 and 4 are swapped/misapplied.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies both the product rule and the chain rule, but is labeled only as "chain". The step changes more than one thing at once, violating the single‑rule‑per‑step rule.
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.