   Chapter 3.6, Problem 50E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use logarithmic differentiation to find the derivative of the function.y = (ln x)cos x

To determine

To find: The derivative of y by using logarithmic differentiation.

Explanation

Given:

The function is y=(lnx)cosx.

Result used: Chain Rule

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

Calculation:

Consider y=(lnx)cosx,

Take natural logarithm on both sides,

lny=ln[(lnx)cosx]lny=cosxln(lnx) (Qlnxa=alnx)

Differentiate both sides with respect to x.

ddx(lny)=ddx(cosxln(lnx))

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

ddx(lny)=cosxddx(ln(lnx))+ln(lnx)ddx(cosx)

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

1ydydx=cosxddx(lnu)+ln(lnx)ddx(cosx)=cosx(1u

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