Derivative of \( \displaystyle - x^{2} + 2 x \)
Problem 2.2171 · medium Mental math
Differentiate \( \displaystyle f(x) = - x^{2} + 2 x \).
- \[ \frac{d}{d x} \left(- x^{2} + 2 x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} \left(- x^{2}\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x - \frac{d}{d x} x^{2} \]constantApply the constant multiple rule to the first term.✓ Proved
- \[ = 2 - 2 x \]powerApply the power rule to both terms.✓ Proved
Answer \( 2 - 2 x \)
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 |
| 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 3 is labeled 'constant' but applies the constant multiple rule to only one term, leaving the other term untouched; this violates the single-rule-per-step constraint. Step 4 is labeled 'power' but applies the power rule to both terms simultaneously, also violating the single-rule-per-step constraint.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 is labeled 'constant' but applies the constant multiple rule to only one term, leaving the other term untouched; this violates the single-rule-per-step constraint. Step 4 is labeled 'power' but applies the power rule to both terms simultaneously, also violating the single-rule-per-step constraint.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 3 is labeled 'constant' but applies the constant multiple rule (pulling out the -1); the vocabulary requires 'constant-multiple' for this operation. Step 4 is labeled 'power' but differentiates two separate terms simultaneously, violating the one-change-per-step constraint.
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 2026-10-13 with SymPy 1.14.0.