Derivative of \( \displaystyle - x^{2} - 2 x \)
Problem 2.1784 · 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
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(- x^{2}\right) \]sumApply the sum rule.✓ Proved
- \[ = - \frac{d}{d x} 2 x - \frac{d}{d x} x^{2} \]constantFactor out the negative sign.✓ Proved
- \[ = - 2 x - 2 \]powerApply the power rule to each term.✓ Proved
Answer \( - 2 x - 2 \)
✓ 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 |
| 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: fail (style) 2026-10-07 — Step 4 is labeled "power" but applies the power rule to the first term and the constant‑multiple rule to the second term. The label should reflect the correct rule for each term.qwen3.6:27b-mlx: fail (style) 2026-10-07 — Step 3 applies the constant-multiple rule to both terms simultaneously, violating the one-change-per-step constraint. Step 4 applies the power rule to both terms simultaneously, also violating the 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 2026-10-07 with SymPy 1.14.0.