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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find dy/dx by implicit differentiation.

12. cos(xy) = 1 + sin y

To determine

To find: The derivative dydx by implicit differentiation.

Explanation

Given:

The equation cos(xy)=1+siny.

Derivative rules:

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

dydx=dydududx.

(2) Product rule: ddx(fg)=fddx(g)+gddx(f)

Calculation:

Obtain the derivative of cos(xy)=1+siny implicit with respect to x.

cos(xy)=1+siny

Differentiate with respect to x on both sides,

ddx(cos(xy))=ddx(1+siny)ddx(cos(xy))=ddx(1)+ddx(siny)ddx(cos(xy))=ddx(siny)

Apply the chain rule (1) and simplify the terms,

ddx(cos(xy))=[ddy(siny)dydx]=cosydydx

Let u=xy and apply the chain rule,

[ddx(cosu)]=cosydydx[ddu(cosu)dudx]=cosydydxsinududx=cosydydx

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