Derivative of \( \displaystyle x \left(x + 2\right) \)
Problem 2.452 · 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
- \[ = \frac{d}{d x} \left(x^{2} + 2 x\right) \]algebraExpand the expression first.✓ Proved
- \[ = \frac{d}{d x} 2 x + \frac{d}{d x} x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 2 x + \frac{d}{d x} 2 x \]powerDifferentiate the first term using the power rule.✓ Proved
- \[ = 2 x + 2 \]constant-multipleDifferentiate the second term by pulling out the constant.✓ 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 |
| 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 (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly expands the function before differentiating, applying one rule per step with accurate labels and notes.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 applies two rules at once: it differentiates x (derivative/power) and pulls out the constant 2 (constant-multiple). The step should be split or labeled to reflect that differentiation occurred, not just constant extraction.gpt-oss:20b: pass 2026-09-20
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.