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Use logarithmic differentiation to find derivative

f(x)=(secx)x

Short Answer

Expert verified

The answer of this question is

f(x)=(ln(secx)+xtanx)(secx)x

Step by step solution

01

Step 1.Given information

We have been given to find the derivative of function

f(x)=(secx)x

02

Step 2.Derivative of given function 

Product rule

(fg)(x)=f(x)g(x)+f(x)g(x)

Quotient rule

fg(x)=f(x)g(x)f(x)g(x)(g(x))2

03

Step 3.Using logarithmic approach 

Taking log both side and then approaching

f(x)    =(secx)xlnf(x)    =ln(secx)x[Taking log in both sides]lnf(x)    =xln(secx)f(x)f(x)    =ddx(xln(secx))Differentiate both sideswith respect toxf(x)f(x)    =ddx(x)ln(secx)+xddxln(secx)    =ln(secx)+xsecxddxsecxUsing product ruleddxlnx=1x    =ln(secx)+xsecx(secxtanx)f(x)    =ddxln(secx=secxtanx]

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