Home›Calculus 1›Product rule›Problem 2.85

Derivative of \( \displaystyle x \left(5 x + 2\right) \)

Problem 2.85 · easy Video

Differentiate \( \displaystyle f(x) = x \left(5 x + 2\right) \).

\[ \frac{d}{d x} x \left(5 x + 2\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} 2 x + \frac{d}{d x} 5 x^{2} \]
sum algebraApply the sum rule. Distribute x into the parentheses.✓ Proved
\[ = 2 \frac{d}{d x} x + 5 \frac{d}{d x} x^{2} \]
constant-multipleFactor out the constants.✓ Proved
\[ = 10 x + 2 \]
power algebraApply the power rule to each term. Multiply the terms.✓ Proved

Answer \( 10 x + 2 \)

Top: the graph of f = x·(5x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 10x + 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. A reviewer's objection to the explanation stands; see the receipt. 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-04: fail — Line 2 distributes x over (5x+2) and applies the sum rule in one step, but is labeled only 'sum'. Line 3's note says 'Distribute x into the parentheses', yet distribution already happened in line 2 and line 3 only simplifies x*(5x) to 5x**2. The math is correct and the answer is right, but the step structure and notes would mislead a student.

Every verdict on record (13)

claude-sonnet-5-5: fail (misleading) 2026-10-04 — Line 2 distributes x over (5x+2) and applies the sum rule in one step, but is labeled only 'sum'. Line 3's note says 'Distribute x into the parentheses', yet distribution already happened in line 2 and line 3 only simplifies x*(5x) to 5x**2. The math is correct and the answer is right, but the step structure and notes would mislead a student.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 2 incorrectly applies the sum rule to a product; the derivative of x*(5*x+2) should use the product rule, not split into Derivative(x*(5*x)) + Derivative(2*x).
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the sum rule to the product x*(5*x+2); a product rule is required. The subsequent steps are based on this incorrect decomposition.
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the sum rule, algebraic simplification, constant multiple rule, and power rule in distinct steps. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Step 2 incorrectly applies the sum rule to a product; the derivative of x*(5*x + 2) should use the product rule, not be split into two separate derivatives. Subsequent steps propagate this error.
gpt-oss:20b: fail 2026-09-17 — Step 2 applies both distribution (x*(5*x+2)=x*(5*x)+2*x) and the sum rule in one line, yet it is labeled only as "sum". This mixes two rules and misleads the student.
deepseek-r1:70b: pass 2026-09-17

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.