Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(4 x - 2 \right)}}{8} \)
Problem 2.611 · hard Video
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. 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 |
| 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: passqwen3.6:27b-mlx: fail (style) — Step 5 applies the chain rule to the inner function (4*x - 2) but is labeled 'algebra'. The differentiation of the inner function should be labeled 'derivative' or 'chain', not 'algebra'. Additionally, Step 7 labels the evaluation of Derivative(x, x) as 'derivative', which is correct, but Step 5's label is incorrect for the operation performed (differentiating the inner term).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 5 applies the chain rule to the inner function (4*x - 2) but is labeled 'algebra'. The differentiation of the inner function should be labeled 'derivative' or 'chain', not 'algebra'. Additionally, Step 7 labels the evaluation of Derivative(x, x) as 'derivative', which is correct, but Step 5's label is incorrect for the operation performed (differentiating the inner term).gpt-oss:20b: pass 2026-09-26qwen3.6:27b-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.