Answer
The proof is below.
Work Step by Step
Let $ y = x^m$. Thus, $y^{\frac{1}{m}}= x$
Taking the derivative on both sides, we find:
$\frac{1}{m}y^{\frac{1-m}{m}}\frac{dy}{dx}= 1$
$\frac{dy}{dx} = my^{\frac{m-1}{m}} = m (x^m)^{\frac{m-1}{m}} = mx^{m-1}$
Thus, we have proven $\frac{d}{dx}(x^m) = mx^{m-1}$ using implicit differentiation.
