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Derivative of \( \displaystyle 2 x \left(x + 1\right) \)

Problem 2.522 · medium Video

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 \( 4 x + 2 \)

Top: the graph of f = 2x·(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4x + 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. How we verify

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

Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line was proved equivalent by SymPy, and the labels name rules that were actually applied. Line 3 combines the sum and constant-multiple rules, but the label 'sum' is acceptable under the contract.

gpt-oss:20b: dismiss — Line 3 splits the sum and pulls out the constant coefficients, which is the standard linearity step, and 'sum' is a legitimate label for it, so this is not a real defect.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-09 — Every line was proved equivalent by SymPy, and the labels name rules that were actually applied. Line 3 combines the sum and constant-multiple rules, but the label 'sum' is acceptable under the contract.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-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 2026-09-20
gpt-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 2026-09-20
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.