Derivatives

2.2 Power Rule

Statement

Theorem

For any integer $n$,

$$ \frac{d}{dx}x^n = nx^{n-1} $$

Example Calculations

  • $\frac{d}{dx}(x^5)=5x^4$
  • $\frac{d}{dx}(x^{-3})=-3x^{-4}$
  • $\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}$ for $x>0$

Proof

For natural $n$, apply the binomial theorem to $(x+h)^n$ and simplify the difference quotient.

Practice

Exercise

Differentiate each function:

  1. $f(x)=7x^4-2x+9$
  2. $g(x)=x^{-5}+3x^2$
  3. $h(x)=\sqrt[3]{x}=x^{1/3}$