Derivative of \( \displaystyle x - \frac{\sin{\left(2 x - 2 \right)}}{2} \)
Problem 2.455 · hard
Differentiate \( \displaystyle f(x) = x - \frac{\sin{\left(2 x - 2 \right)}}{2} \).
- \[ \frac{d}{d x} \left(x - \frac{\sin{\left(2 x - 2 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \frac{\sin{\left(2 x - 2 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} x - \frac{\frac{d}{d x} \sin{\left(2 x - 2 \right)}}{2} \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = 1 - \frac{\frac{d}{d x} \sin{\left(2 x - 2 \right)}}{2} \]derivativeThe derivative of x is 1.✓ Proved
- \[ = - \frac{\cos{\left(2 x - 2 \right)} \frac{d}{d x} \left(2 x - 2\right)}{2} + 1 \]chainApply the chain rule to the sine function.✓ Proved
- \[ = 1 - \cos{\left(2 x - 2 \right)} \]derivative algebraThe derivative of 2*x - 2 is 2. Simplify the expression by multiplying the constants.✓ Proved
Answer \( 1 - \cos{\left(2 x - 2 \right)} \)
✓ 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 |
| 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: pass — The solution correctly applies the sum, constant-multiple, chain, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the sum, constant-multiple, chain, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the sum, constant-multiple, derivative, and chain rules in separate steps. The labels are accurate and the logic is sound.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the sum rule, constant multiple rule, chain rule, and basic derivatives in a logical sequence. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20
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.