Revisiting Power Rule for fractional exponents

Suppose $n$ is any positive integer for which $y=x^{\frac{1}{n}}$ makes sense. Earlier we had found $\frac{dy}{dx}$ using the definition of the derivative and that the power rule is true for positive integers.

Here we will find the same using implicit differentiation and that the power rule is true for positive integers.

$y=x^{\frac{1}{n}}$ implies $x=y^n$. Now we differentiate both sides with respect to $x$ and get

$1=n{y}^{n-1}\frac{dy}{dx}$

or, $\frac{dy}{dx}=\frac{1}{n{y}^{n-1}}=\frac{1}{n}\frac{y}{y^n}=\frac{1}{n}\frac{x^{\frac{1}{n}}}{x}=\frac{1}{n}{x}^{\frac{1}{n}-1}$