Derivatives and the power rule

A derivative is the slope of a curve at a point. The power rule lets you differentiate any polynomial in one line.

1 分で読めます2026年10月18日更新執筆：Sumzo teamレビュー：Sumzo team

A straight line has the same slope everywhere, but a curve does not. The derivative $f'(x)$ gives the slope of the curve $y = f(x)$ at each value of $x$ — the instantaneous rate of change.

The power rule

\frac{d}{dx}\left(a x^n\right) = n a x^{n-1}

Bring the power down and reduce it by one.

Differentiate a sum term by term.
A constant term has derivative 0 (a flat line has no slope).
The derivative of $ax$ is $a$ .

解き方付き例題

Differentiate $3x^3 - 2x + 5$

Find $f'(x)$ and $f''(x)$ for $f(x) = 3x^3 - 2x + 5$ .

解き方を表示

Apply the power rule to each term. $f'(x) = 9x^2 - 2$
Differentiate again for the second derivative. $f''(x) = 18x$

答え： $f'(x) = 9x^2 - 2$ and $f''(x) = 18x$

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解き方付き例題

Expand first, then differentiate

Differentiate $x^2(x + 1)$ .

解き方を表示

Expand first so every term is a power of $x$ . $x^2(x + 1) = x^3 + x^2$
Apply the power rule. $\frac{d}{dx}(x^3 + x^2) = 3x^2 + 2x$

答え： $3x^2 + 2x$

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