Derivative of \( \displaystyle x \left(- x - 2\right) \)
Problem 2.1400 · medium
Differentiate \( \displaystyle f(x) = x \left(- x - 2\right) \).
- \[ \frac{d}{d x} x \left(- x - 2\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(- x - 2\right) + \left(- x - 2\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(- x - 2\right) - x - 2 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x \frac{d}{d x} \left(-2\right) + x \frac{d}{d x} \left(- x\right) - x - 2 \]sumApply the sum rule to the second part.✓ Proved
- \[ = - 2 x - 2 \]derivative constant algebra simplifyDifferentiate the terms in the sum. The derivative of the constant -2 is 0. Distribute the 1. Combine like terms.✓ 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 |
| 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 |
| 8 | ✓ 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 product rule, sum rule, and basic derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the product rule, sum rule, and basic derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the product rule, sum rule, and basic 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 2026-10-03
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 2026-10-03 with SymPy 1.14.0.