Derivative of \( \displaystyle \frac{\left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \)
Problem 2.1109 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \).
- \[ \frac{d}{d x} \frac{\left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\sin{\left(4 x + 2 \right)} - \cos{\left(4 x + 2 \right)}\right) e^{4 x + 2}}{8} \]algebraDistribute the terms inside the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} e^{4 x + 2} \sin{\left(4 x + 2 \right)}}{8} - \frac{\frac{d}{d x} e^{4 x + 2} \cos{\left(4 x + 2 \right)}}{8} \]sumApply the difference rule for derivatives.✓ Proved
- \[ = - \frac{e^{4 x + 2} \frac{d}{d x} \cos{\left(4 x + 2 \right)}}{8} - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} e^{4 x + 2}}{8} + \frac{\frac{d}{d x} e^{4 x + 2} \sin{\left(4 x + 2 \right)}}{8} \]productApply the product rule to the second term.✓ Proved
- \[ = \frac{e^{4 x + 2} \frac{d}{d x} \sin{\left(4 x + 2 \right)}}{8} - \frac{e^{4 x + 2} \frac{d}{d x} \cos{\left(4 x + 2 \right)}}{8} + \frac{\sin{\left(4 x + 2 \right)} \frac{d}{d x} e^{4 x + 2}}{8} - \frac{\cos{\left(4 x + 2 \right)} \frac{d}{d x} e^{4 x + 2}}{8} \]productApply the product rule to the first term.✓ Proved
- \[ = e^{4 x + 2} \sin{\left(4 x + 2 \right)} \]chain algebra simplify simplifyApply the chain rule to the trigonometric and exponential functions. Distribute the negative sign and group terms. Combine like terms. Simplify the final expression.✓ Proved
Answer \( e^{4 x + 2} \sin{\left(4 x + 2 \right)} \)
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 |
| 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 6 applies the chain rule to several functions at once, violating the rule that each step must change only one thing. The step should be split into separate applications of the chain rule for sin, cos, and exp.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: fail (error) 2026-09-28 — Step 6 applies the chain rule to several functions at once, violating the rule that each step must change only one thing. The step should be split into separate applications of the chain rule for sin, cos, and exp.qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.