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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Calculate y'.

28. y = (cos x)x

To determine

To calculate: The derivative of y.

Explanation

Given:

The function is y=(cosx)x.

Result used: Chain Rule

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

Calculation:

Consider y=(cosx)x.

Take natural logarithm on both sides,

lny=ln[(cosx)x]lny=xlncosx     (lnxa=alnx)

Differentiate both sides with respect to x.

ddx(lny)=ddx(xlncosx)

Apply the product rule ddx(fg)=fddx(g)+gddx(f),

1ydydx=xddx(ln(cosx))+ln(cosx)ddx(x)

Let u=cosx and apply the chain rule as stated above,

1ydydx=xddx(lnu)+ln(cosx)(1x11)=x(1ududx

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