Derivative of \( \displaystyle 2 x \left(x + 1\right) \)
Problem 2.522 · medium
Differentiate \( \displaystyle f(x) = 2 x \left(x + 1\right) \).
- \[ \frac{d}{d x} 2 x \left(x + 1\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(2 x^{2} + 2 x\right) \]algebraExpand the expression.✓ Proved
- \[ = 2 \frac{d}{d x} x + 2 \frac{d}{d x} x^{2} \]sumApply the sum rule.✓ Proved
- \[ = 4 x + 2 \]power algebraDifferentiate each term using the power rule. Simplify the expression.✓ Proved
Answer \( 4 x + 2 \)
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 |
| 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: fail (error) — Step 3 applies both the sum rule and the constant‑multiple rule, but only labels "sum". Each step must change only one thing and name the exact rule used.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies both the sum rule and the constant‑multiple rule, but only labels "sum". Each step must change only one thing and name the exact rule used.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 1 uses an unlabelled rule and does not apply the product rule; it incorrectly treats the derivative of the product as a single operation, violating the rule‑granularity requirement.qwen3.6:27b-mlx: pass 2026-09-20gpt-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.