Derivative of \( \displaystyle x \left(2 - x\right) \)
Problem 2.679 · medium
Differentiate \( \displaystyle f(x) = x \left(2 - x\right) \).
- \[ \frac{d}{d x} x \left(2 - x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x^{2} + 2 x\right) \]algebraExpand the expression.✓ Proved
- \[ = \frac{d}{d x} 2 x - \frac{d}{d x} x^{2} \]sumApply the difference rule.✓ Proved
- \[ = 2 - \frac{d}{d x} x^{2} \]constantThe derivative of 2*x is 2.✓ Proved
- \[ = 2 - 2 x \]powerApply the power rule to x**2.✓ Proved
Answer \( 2 - 2 x \)
✓ 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 |
| 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 (style) — Step 4 labels the derivative of 2*x as "constant"; the correct rule name from the vocabulary is "constant-multiple". The step itself is correct, but the label does not match the rule applied.qwen3.6:27b-mlx: pass — The solution correctly expands the function, applies the sum/difference rule, and computes the derivatives of the individual terms using standard rules. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly expands the function, applies the sum/difference rule, and computes the derivatives of the individual terms using standard rules. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (style) 2026-09-21 — Step 4 labels the derivative of 2*x as "constant"; the correct rule name from the vocabulary is "constant-multiple". The step itself is correct, but the label does not match the rule applied.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (style) 2026-09-21 — Step 4 labels the derivative of 2*x as "constant", but the correct rule is "constant-multiple". The label does not match the rule actually applied.
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.