9 - Power Rule for Differentiation (6)

Suppose $y=f(x)=x^q$  where $q$  is an irrational number. And, suppose ${\left\{r_k\right\}}_{k=1}^{k=\infty }$  is an infinite sequence of rational numbers such that $\underset{k\to \infty }{\lim }{r}_k=q$. Then

$\frac{dy}{dx}=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$=\underset{h\to 0}{\lim }\frac{{(x+h)}^q-x^q}{h}$

$=\underset{h\to 0}{\lim }\frac{{(x+h)}^{\underset{k\to \infty }{\lim }{r}_k}-x^{\underset{k\to \infty }{\lim }{r}_k}}{h}$

$=\underset{k\to \infty }{\lim }\underset{h\to 0}{\lim }\frac{{(x+h)}^{r_k}-x^{r_k}}{h}$

$=\underset{k\to \infty }{\lim }{r}_k{x}^{r_k-1}$

$=\underset{k\to \infty }{\lim }{r}_k\underset{k\to \infty }{\lim }{x}^{r_k-1}$

$=q{x}^{q-1}$