Let $y={x}^{\frac{m}{n}}$ where $n$ is a postive integer.

Note that any fraction $\frac{a}{b}$ can be written as $\frac{m}{n}$ where $n$ is a postive integer.

Earlier we had fouind the derivative of $y={x}^{\frac{m}{n}}$ using the definition of the derivative and the fact that the power rule is true for both positive and integer exponents and for fractional exponents of the form $\frac{1}{n}$ where $n$ is a postive integer. Here we will show the same except that instead of using the definition of the derivative, we will use the implicit differentiation.

$y={x}^{\frac{m}{n}}={({x}^{m})}^{\frac{1}{n}}$

Differentiating with respect to $x$, we get

$y=\frac{1}{n}{({x}^{m})}^{\frac{1}{n}-1}\cdot m{x}^{m-1}=\frac{m}{n}{x}^{\frac{m}{n}-m}\cdot {x}^{m-1}=\frac{m}{n}{x}^{\frac{m}{n}-m+m-1}=\frac{m}{n}{x}^{\frac{m}{n}-1}$

:-)