Derivative of \( \displaystyle \frac{x}{2} - \frac{\sin{\left(2 x + 2 \right)}}{4} \)
Problem 2.1451 · 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
- \[ = - \frac{\cos{\left(2 x + 2 \right)} \frac{d}{d x} x}{2} + \frac{1}{2} \]derivativeDifferentiate the first term.✓ Proved
- \[ = \frac{1}{2} - \frac{\cos{\left(2 x + 2 \right)}}{2} \]derivative simplifyDifferentiate the remaining x term. 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 |
| 10 | ✓ 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 applies the sum rule to the inner function 2*x+2 but omits the derivative of the constant term 2. The correct derivative is 2*Derivative(x,x)+Derivative(2,x)=2*1+0, so the constant 2 should not be dropped. This makes the subsequent steps incorrect.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 applies the sum rule to the inner function 2*x+2 but omits the derivative of the constant term 2. The correct derivative is 2*Derivative(x,x)+Derivative(2,x)=2*1+0, so the constant 2 should not be dropped. This makes the subsequent steps incorrect.qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.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.