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The Power Rule can be proved using implicit differentiation for the case where $ n $ in a rational number, $ n = p/q, $ and $ y = f(x) = x" $ 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' = \frac {p}{q} x^{(p/q)-1} $

$$

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

$$

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Campbell University

Harvey Mudd College

University of Nottingham

Boston College

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