6 - Power Rule for Differentiation (5)

Let $y=f(x)=x^{\frac{m}{n}}$ ,where $m$ is a positive integer and $n$ is any integer. Then

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

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

$=\underset{h\to 0}{\lim }\frac{{({(x+h)}^{\frac{1}{n}})}^m-{(x^{\frac{1}{n}})}^m}{h}$

$=\underset{h\to 0}{\lim }\frac{[{(x+h)}^{\frac{1}{n}}-x^{\frac{1}{n}}][{({(x+h)}^{\frac{1}{n}})}^{m-1}+{({(x+h)}^{\frac{1}{n}})}^{m-2}{x}^{\frac{1}{n}}+\cdot \cdot \cdot +{(x+h)}^{\frac{1}{n}}{(x^{\frac{1}{n}})}^{m-2}+{(x^{\frac{1}{n}})}^{m-1}]}{h}$

$=\underset{h\to 0}{\lim }\frac{(x+h{)}^{\frac{1}{n}}-x^{\frac{1}{n}}}{h}[{({(x+h)}^{\frac{1}{n}})}^{m-1}+{({(x+h)}^{\frac{1}{n}})}^{m-2}{x}^{\frac{1}{n}}+\cdot \cdot \cdot +{(x+h)}^{\frac{1}{n}}{(x^{\frac{1}{n}})}^{m-2}+{(x^{\frac{1}{n}})}^{m-1}]$

$=\frac{1}{n}{x}^{\frac{1}{n}-1}[m{x}^{\frac{m-1}{n}}]$

$=\frac{m}{n}{x}^{\frac{m}{n}-1}$