Derivative of \( \displaystyle x \left(2 - x\right) \)
Problem 2.679 · medium Video
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. How we verify
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: passqwen3.6:27b-mlx: fail (style) — Step 4 applies the constant-multiple rule (derivative of 2x is 2*1) but labels it 'constant', which typically refers to the derivative of a constant being zero. Step 5 applies the power rule but labels it 'power' while the note says 'Apply the power rule to x**2', which is acceptable, but the previous step's label is imprecise given the vocabulary constraints where 'constant-multiple' exists.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 4 applies the constant-multiple rule (derivative of 2x is 2*1) but labels it 'constant', which typically refers to the derivative of a constant being zero. Step 5 applies the power rule but labels it 'power' while the note says 'Apply the power rule to x**2', which is acceptable, but the previous step's label is imprecise given the vocabulary constraints where 'constant-multiple' exists.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-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.