Chapter 2.3, Problem 84ES

The purpose of this exercise is to extend the power rule (Theorem $\mathrm{2.3.2}$ ) to any integer exponent. Let $f\left(x\right)=x^n$ , where $n$ is any integer. If $n>0$ , $f^{\prime }\left(x\right)=n{x}^{n-1}$ by Theorem $\mathrm{2.3.2}$ .

(a) Show that the conclusion of Theorem $\mathrm{2.3.2}$ holds in the case $n=0$ .

(b) Suppose that $n<0$ and set $m=-n$ so that

$f\left(x\right)=x^n=x^{-m}=\frac{1}{x^m}$

Use Definition $\mathrm{2.2.1}$ and Theorem $\mathrm{2.3.2}$ to show that

$\frac{d}{dx}\left[\frac{1}{x^m}\right]=-m{x}^{m-1}\cdot \frac{1}{x^{2m}}$

and conclude that $f^{\prime }\left(x\right)=n{x}^{n-1}$ .