Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8} \)
Problem 2.611 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(4 x - 2 \right)}}{8} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(4 x - 2 \right)}}{8} \]constant-multipleFactor out the constant 1/8.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 2 \right)} \frac{d}{d x} \left(4 x - 2\right)}{8} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]algebra algebraSimplify the constant multiplication. Simplify 4/8 to 1/2.✓ Proved
- \[ = - \frac{\cos{\left(4 x - 2 \right)}}{2} + \frac{d}{d x} \frac{x}{2} \]derivativeDifferentiate the remaining x term.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(4 x - 2 \right)}}{2} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(4 x - 2 \right)}}{2} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 |
| 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 5 applies two rules (constant‑multiple and derivative) but is labeled only as "algebra". It should be split into a constant‑multiple step followed by a derivative step, or labeled appropriately.qwen3.6:27b-mlx: pass — The solution correctly applies the sum, constant-multiple, and chain rules in separate steps. The algebraic simplifications and final derivative evaluations are accurate and properly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum, constant-multiple, and chain rules in separate steps. The algebraic simplifications and final derivative evaluations are accurate and properly labeled.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 applies two rules (constant‑multiple and derivative) but is labeled only as "algebra". It should be split into a constant‑multiple step followed by a derivative step, or labeled appropriately.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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 2026-09-26 with SymPy 1.14.0.