The algebraic computation

On your homework you have looked at several special cases of the following computation.

In what follows, p > 1 is an integer, $a\neq 0$ and $a+h\neq 0$. If p is even, we also assume a > 0 and a+h > 0. As usual, we reduce the problem to factoring a difference of postive integer powers. This time the pattern looks like

$$\frac{A-B}{A^p - B^p} = \frac{1}{A^{p-1} + A^{p-2}B + \cdots + AB^{p-2} + B^{p-1}}.$$

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

from which we see that

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

which shows that the power rule extends to functions defined on intervals whose rules are given by $f(x) = \sqrt[p]{x}$.