Chapter 3.3, Problem 94E

Looking ahead: Power Rule for negative integers Suppose n is a negative integer and f(x) = xⁿ. Use the following steps to prove that f′ (a) = na^n−l, which means the Power Rule for positive integers extends to all integers. This result is proved in Section 3.4 by a different method.

1. a. Assume that m = − n, so that m > 0. Use the definition $f^{\prime }(a)=\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}=\underset{x\to a}{\lim }\frac{x^{-m}-a^{-m}}{x-a}$. Simplify using the factoring rule (which is valid for n > 0) $x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+\cdots +x{a}^{n-2}+a^{n-1})$ until it is possible to take the limit.
2. b. Use this result to find $\frac{d}{dx}(x^{-7})$ and $\frac{d}{dx}\left(\frac{1}{x^{10}}\right)$.