Problem 48E

48. The Power Rule can be proved using implicit differentiation for the case where $n$ is a rational number, $n=p / q$, and $y=f(x)=x^{n}$ is assumed beforehand to be a differentiable function. If $y=x^{p / q}$, then $y^{q}=x^{p}$. Use implicit differentiation to show that

$$ y^{\prime}=\frac{p}{q} x^{(p / q)-1} $$

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