A note on the power rule

We have proved the power rule for integer powers by a direct appeal to the definition of derivative, and for roots by appeal to the rule for differentiating inverses. To complete the derivation, if p is a rational number, write p = N/D where N and D are integers, and compute with the chain rule:

$$(x^p)' = \left(x^{N/D}\right)' = \left(\left(x^{1/D}\right)^... ...{N-1}\frac{1}{D}x^{(1/D)-1} = \frac{N}{D}x^{(N/D)-1} = px^{p-1}$$

For exponents which are real numbers, say $r\neq 0$, x^r = 0 if x = 0 and for $x \neq 0$ we define $x^r = e^{r\ln(x)}$. Then for x > 0 we can write

$$\left(x^r\right)' = \left(e^{r\ln(x)}\right))' = \left(e^{r\ln(x)}\right)r\frac{1}{x} = rx^rx^{-1} = rx^{r-1}$$

It is too difficult at this time to discuss the behavior of the derivative at x = 0.