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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.

19. sin(xy) = cos(x + y)

To determine

To find: The derivative dydx by implicit differentiation.

Explanation

Given:

The equation sin(xy)=cos(x+y).

Derivative rules:

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

dydx=dydududx.

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

ddx[f(x)g(x)]=f(x)ddx(g(x))+g(x)ddxf(x).

Calculation:

Obtain the derivative of sin(xy)=cos(x+y) implicitly with respect to x.

sin(xy)=cos(x+y)

Differentiate with respect to x on both sides,

ddx(sin(xy))=ddx(cos(x+y))

Let u=xy and k=x+y.

ddx(sinu)=ddx(cosk)

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

ddu(sinu)dudx=ddk(cosk)dkdxcosududx=sinkdkdx

Substitute the value u=xy and k=x+y,

cos(xy)ddx(xy)=sin(x+y)ddx(x+y)cos(xy)

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