Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(2 x - 2 \right)}}{4} \)
Problem 2.1399 · hard
Differentiate \( \displaystyle f(x) = \frac{x}{2} - \frac{\sin{\left(2 x - 2 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x - 2 \right)}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{d}{d x} \frac{\sin{\left(2 x - 2 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} - \frac{\frac{d}{d x} \sin{\left(2 x - 2 \right)}}{4} \]constant-multipleFactor out the constant 1/4.✓ Proved
- \[ = - \frac{\cos{\left(2 x - 2 \right)} \frac{d}{d x} \left(2 x - 2\right)}{4} + \frac{d}{d x} \frac{x}{2} \]chainApply the chain rule to the sine term.✓ Proved
- \[ = - \frac{\cos{\left(2 x - 2 \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \frac{x}{2} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = - \frac{\cos{\left(2 x - 2 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \frac{x}{2} \]constant-multiple algebraFactor out the constant 2. Simplify the coefficients.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(2 x - 2 \right)}}{2} \]derivative simplifyDifferentiate the remaining terms. Simplify the final expression.✓ Proved
Answer \( \frac{1}{2} - \frac{\cos{\left(2 x - 2 \right)}}{2} \)
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 5 incorrectly drops the derivative of the constant term – it applies the sum rule but omits the zero contribution from Derivative(-2, x).qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (error) 2026-10-03 — Step 5 incorrectly drops the derivative of the constant term – it applies the sum rule but omits the zero contribution from Derivative(-2, x).qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 claims to apply the 'sum' rule to the inner function `2*x - 2`, but it only removes the constant term `-2` without showing the derivative of that constant term (which is 0). The step effectively applies the constant rule and sum rule simultaneously or skips the evaluation of the constant's derivative, violating the one-rule-per-step constraint and logical completeness.gpt-oss:20b: fail (error) 2026-10-03 — Step 5 applies the sum rule to the inner function 2*x-2 but omits the derivative of the constant term 2 (which is 0). This is a missing term and constitutes a mistake in the derivation.
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.