Derivative of \( \displaystyle - \frac{x}{2} + \frac{\sin{\left(2 x + 4 \right)}}{4} \)
Problem 2.1056 · hard
Differentiate \( \displaystyle f(x) = - \frac{x}{2} + \frac{\sin{\left(2 x + 4 \right)}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{x}{2} + \frac{\sin{\left(2 x + 4 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{d}{d x} \frac{\sin{\left(2 x + 4 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{x}{2}\right) + \frac{\frac{d}{d x} \sin{\left(2 x + 4 \right)}}{4} \]constantFactor out the constant coefficient.✓ Proved
- \[ = \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} \left(2 x + 4\right)}{4} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]chainApply the chain rule to the sine function.✓ Proved
- \[ = \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} 2 x}{4} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]constantDifferentiate the constant term inside the argument.✓ Proved
- \[ = \frac{\cos{\left(2 x + 4 \right)} \frac{d}{d x} x}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]constant-multiple algebra simplifyFactor out the constant 2. Multiply the constants. Simplify the fraction.✓ Proved
- \[ = \frac{\cos{\left(2 x + 4 \right)}}{2} + \frac{d}{d x} \left(- \frac{x}{2}\right) \]derivativeEvaluate the derivative of x.✓ Proved
- \[ = \frac{\cos{\left(2 x + 4 \right)}}{2} - \frac{1}{2} \]derivative simplifyEvaluate the derivative of -x/2. Final simplification.✓ Proved
Answer \( \frac{\cos{\left(2 x + 4 \right)} - 1}{2} \)
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 |
| 11 | ✓ 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 (error) — Step 5 is mathematically incorrect. The derivative of a sum is the sum of the derivatives, so Derivative(2*x + 4, x) becomes Derivative(2*x, x) + Derivative(4, x). The step incorrectly drops the derivative of the constant term 4 (which is 0) without showing it, effectively applying the sum rule and the constant rule simultaneously, violating the one-rule-per-step constraint. Furthermore, the label 'constant' is inappropriate for differentiating a sum.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 5 is mathematically incorrect. The derivative of a sum is the sum of the derivatives, so Derivative(2*x + 4, x) becomes Derivative(2*x, x) + Derivative(4, x). The step incorrectly drops the derivative of the constant term 4 (which is 0) without showing it, effectively applying the sum rule and the constant rule simultaneously, violating the one-rule-per-step constraint. Furthermore, the label 'constant' is inappropriate for differentiating a sum.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 5 is mathematically incorrect; it replaces Derivative(2*x + 4, x) with Derivative(2*x, x), effectively dropping the constant term 4 from the differentiation process without justification. The derivative of a sum is the sum of the derivatives, so Derivative(2*x + 4, x) should become Derivative(2*x, x) + Derivative(4, x), not just Derivative(2*x, x).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.