   Chapter 3.6, Problem 40E

Chapter
Section
Textbook Problem

Use logarithmic differentiation to find the derivative of the function. y = e − x cos 2 x x 2 + x + 1

To determine

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

Explanation

Given:

The function is y=excos2xx2+x+1.

Result used:

Chain Rule:

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

Calculation:

Consider y=excos2xx2+x+1.

Take natural logarithm on both sides,

lny=ln[excos2xx2+x+1]lny=ln(excos2x)ln(x2+x+1)         (Qlnab=lnalnb)lny=lnex+lncos2xln(x2+x+1)      (Qlnab=lna+lnb)lny=x+2lncosxln(x2+x+1)         (Qlnxa=alnxlne=1)

Differentiate both sides with respect to x.

ddx(lny)=ddx(x+2lncosxln(x2+x+1))ddx(lny)=ddx(x)+ddx(2lncosx)ddx(ln(x2+x+1))

Let u=x2+x+1 and v=cosx.

ddx(lny)=ddx(x)+ddx(2lnv)ddx(lnu)=ddx(x)+2ddx(lnv)ddx(lnu)

Use the chain rule stated above,

1ydydx=(1x11)+2(1vdvdx

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