Home›Calculus 1›Power rule›Problem 2.2289

Derivative of \( \displaystyle 12 x^{3} - 18 x^{2} + 9 x \)

Problem 2.2289 · medium Video

Differentiate \( \displaystyle f(x) = 12 x^{3} - 18 x^{2} + 9 x \).

\[ \frac{d}{d x} \left(12 x^{3} - 18 x^{2} + 9 x\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} 9 x - \frac{d}{d x} 18 x^{2} + \frac{d}{d x} 12 x^{3} \]
sumApply the sum rule to differentiate each term separately.✓ Proved
\[ = 9 \frac{d}{d x} x - 18 \frac{d}{d x} x^{2} + 12 \frac{d}{d x} x^{3} \]
constant-multiplePull out the constant coefficients.✓ Proved
\[ = 36 x^{2} - 36 x + 9 \]
power algebraApply the power rule to each term. Multiply the constants.✓ Proved

Answer \( 9 \left(2 x - 1\right)^{2} \)

Top: the graph of f = 12x³ - 18x² + 9x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 36x² - 36x + 9.

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the sum, constant-multiple, and power rules in sequence. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the sum, constant-multiple, and power rules in sequence. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the sum, constant-multiple, and power rules in separate steps. The final simplification to the factored form is algebraically valid.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the sum, constant-multiple, and power rules in sequence. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.