AP Calculus BC / Unit 2: Differentiation: Definition and Fundamental Properties
2.5 Applying the Power Rule
Notes
$$\frac{d}{dx}x^n=nx^{n-1}$$
- Works for any real $n$: rewrite roots and fractions as powers first.
- $\displaystyle \sqrt{x}=x^{1/2}$, $\displaystyle \frac{1}{x^3}=x^{-3}$.
Practice
$\displaystyle \frac{d}{dx}x^7=$
Bring down the exponent, then subtract one.
$\displaystyle \frac{d}{dx}\frac{1}{x^2}=$
$x^{-2}\to-2x^{-3}$.
$\displaystyle \frac{d}{dx}\sqrt{x}=$
$\displaystyle x^{1/2}\to\frac{1}{2}x^{-1/2}$.
Ready for more? Take the Unit 2 test.