Negative integer powers

We have seen already that if $f:(-\infty,\infty)\rightarrow(-\infty,\infty)$ is given by f(x) = x^p where $p\in\{1, 2, 3, \dots\}$ then f'(x) = px^p-1. We will now extend this to negative integers as well. Suppose that p is a negative integer, so that we can write p = -q, where q is a positive integer. Define $f:(-\infty,0)\cup(0,\infty)$ by f(x) = x^p. Then we have

$$\begin{array} {rcl} {\displaystyle \frac{f(a+h)-f(a)}{h}} & ... ...+h)+\cdots +a(a+h)^{q-2} +(a+h)^{q-1})}{a^q(a+h)^q}}\end{array}$$

Note that the numerator has q summands, each of which has a limit of a^q-1 as h approaches , so we see that

$$\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h} = \frac{-qa^{q-1}}{a^qa^q} = \frac{-q}{a^{q+1}} = -qa^{-q-1} = pa^{p-1}$$

Thus the power rule holds for $p\in\{\pm 1, \pm 2, \cdots\}$.

What is remarkable is that the power rule never turns up a rate of change with the exponent -1, for to do so, the original exponent would be , which denotes a constant function, and the rate of change of any constant function is !