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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find dy/dx by implicit differentiation.

9. x 2 x + y = y 2 + 1

To determine

To find: The derivative dydx by implicit differentiation.

Explanation

Given:

The equation x2x+y=y2+1.

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then

dydx=dydududx.

(2) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2.

Calculation:

Obtain the derivative of x2x+y=y2+1 implicitly with respect to x.

x2x+y=y2+1

Differentiate with respect to x on both sides,

ddx(x2x+y)=ddx(y2+1)(x+y)ddx(x2)(x2)ddx(x+y)(x+y)2=ddx(y2+1)[(x+y)ddx(x2)][(x2)[ddx(x)+dydx]](x+y)2=ddx(y2)+ddx(1)[(x+y)2x][(x2)[1+dydx]](x+y)2=ddx<

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