Home›Calculus 1›Power rule›Problem 2.2286

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

Problem 2.2286 · medium Video

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

\[ \frac{d}{d x} \left(- x^{3} + 3 x^{2} - 3 x\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x^{2} + \frac{d}{d x} \left(- x^{3}\right) \]
sumApply the sum rule to differentiate each term separately.✓ Proved
\[ = - 3 \frac{d}{d x} x + 3 \frac{d}{d x} x^{2} - \frac{d}{d x} x^{3} \]
constant-multiplePull out the constant coefficients.✓ Proved
\[ = - 3 x^{2} + 6 x - 3 \]
power simplifyApply the power rule to each term. Simplify the resulting expression.✓ Proved

Answer \( - 3 \left(x - 1\right)^{2} \)

Top: the graph of f = -x³ + 3x² - 3x. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -3x² + 6x - 3.

✓ 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 separate steps, adhering to the one-change-per-step constraint. The labels are accurate and from the fixed 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 separate steps, adhering to the one-change-per-step constraint. The labels are accurate and from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
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 separate steps, adhering to the one-change-per-step constraint. The labels are accurate and from the allowed 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.