Derivative of \( \displaystyle \frac{\left(5 \sin{\left(4 x + 2 \right)} - 5 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \)
Problem 2.1429 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
- \[ \frac{d}{d x} \frac{\left(5 \sin{\left(4 x + 2 \right)} - 5 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) \frac{d}{d x} e^{4 x + 2}}{8} + \frac{5 e^{4 x + 2} \frac{d}{d x} \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right)}{8} \]productApply the product rule.✓ Proved
- \[ = \frac{5 \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{2} + \frac{5 \left(4 \sin{\left(4 x + 2 \right)} + 4 \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]derivativeDifferentiate the sine and cosine terms using the chain rule.✓ Proved
- \[ = 5 e^{4 x + 2} \sin{\left(4 x + 2 \right)} \]algebra simplify simplifyDistribute the exponential term. Combine like terms. Simplify the final expression.✓ Proved
Answer \( 5 e^{4 x + 2} \sin{\left(4 x + 2 \right)} \)
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 |
| 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: fail (style) — Step 4 applies the chain rule to differentiate the inner functions but is labeled 'derivative', which is imprecise given the available vocabulary includes 'chain'. Furthermore, Step 4 performs two distinct differentiations (of the trigonometric difference and the exponential) in a single step, violating the one-change-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 4 applies the chain rule to differentiate the inner functions but is labeled 'derivative', which is imprecise given the available vocabulary includes 'chain'. Furthermore, Step 4 performs two distinct differentiations (of the trigonometric difference and the exponential) in a single step, violating the one-change-per-step constraint.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 4 is labeled 'derivative' but performs multiple operations: it differentiates the composite trigonometric terms (requiring the chain rule) and the exponential term, and also distributes the results. The label 'derivative' is insufficient for applying the chain rule, and the step violates the one-change-per-step constraint by differentiating multiple sub-expressions simultaneously.gpt-oss:20b: pass 2026-10-03
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.