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

Problem 2.711 · medium Video

Differentiate \( \displaystyle f(x) = 2 x \left(2 x + 1\right) \).

\[ \frac{d}{d x} 2 x \left(2 x + 1\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} 2 x + \frac{d}{d x} 4 x^{2} \]
algebra algebraDistribute the term 2*x into the parentheses. Simplify the terms inside the derivatives.✓ Proved
\[ = 2 \frac{d}{d x} x + 4 \frac{d}{d x} x^{2} \]
constant-multiplePull out the constant factors.✓ Proved
\[ = 8 x + 2 \]
power algebraApply the power rule to each term. Multiply the constants.✓ Proved

Answer \( 8 x + 2 \)

Top: the graph of f = 2x·(2x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 8x + 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
6	✓ 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 2 incorrectly distributes the product inside the derivative, applying two rules at once and violating the product rule. The correct approach is to apply the product rule to 2*x and (2*x+1), not to split the derivative into two separate terms.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-12: pass — Every line is equal to the previous one (SymPy proved each), and each step does one thing with an acceptable label. The step from line 2 to line 3 to line 4 is fine, and the final answer 8*x+2 is correct, since d/dx(4x^2+2x)=8x+2.

gpt-oss:20b: dismiss — Line 2 is a single algebraic distribution of 2*x over (2*x+1), SymPy proved equality, and the product rule is not required (the derivative is split only later, as a sum, which is legitimate).

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — Every line is equal to the previous one (SymPy proved each), and each step does one thing with an acceptable label. The step from line 2 to line 3 to line 4 is fine, and the final answer 8*x+2 is correct, since d/dx(4x^2+2x)=8x+2.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 2 incorrectly distributes the product inside the derivative, applying two rules at once and violating the product rule. The correct approach is to apply the product rule to 2*x and (2*x+1), not to split the derivative into two separate terms.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21

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.