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 Video
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 |
| 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 (style) — Step 3 incorrectly labels the product rule as "chain" and step 4 incorrectly labels the chain rule as "algebra". Each step must apply exactly one rule and use the correct vocabulary label.qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions using the chain rule and derivative rules for sin, cos, and exp. The labels are swapped and misapplied.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'chain' but performs the expansion of the product rule (distributing the derivative over the product), which was already initiated in Step 2. Step 4 is labeled 'algebra' but performs the actual differentiation of the inner functions using the chain rule and derivative rules for sin, cos, and exp. The labels are swapped and misapplied.gpt-oss:20b: fail (style) 2026-10-03 — Step 3 incorrectly labels the product rule as "chain" and step 4 incorrectly labels the chain rule as "algebra". Each step must apply exactly one rule and use the correct vocabulary label.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 2026-10-03 with SymPy 1.14.0.