   Chapter 3.5, Problem 48E

Chapter
Section
Textbook Problem

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) = xn is assumed beforehand to be a differentiable function. If y = xp/q, then yq = xp. Use implicit differential ion to show that y ' = p q x ( p / q ) − 1

To determine

To show: The derivative of the given function is y(x)=pqxpq1 by using implicit differentiation.

Explanation

The derivative of the function is dydx=pqxpq1.

Given:

The given function is y=xn, where n=pq is a rational number.

Derivative rule: Chain rule

If y=f(u) and u=g(x)  are both differentiable function, then dydx=dydududx.

Proof:

Rewrite the given function,

y=xpqyq=xp

Differentiate implicitly with respect to x,

ddx(yq)=ddx(xp)ddx(yq)=pxp1

Apply the chain rule and simplify the terms,

ddy(yq)dydx=pxp1qyq1dydx=pxp1<

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