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Use implicit differentiation and the fact that secsec-1x=xfor all xin the domain of sec-1xto prove that ddxsec-1x=1xx2-1. You will have to consider the casesx>1andx<-1 separately.

Short Answer

Expert verified

We proved ddxsec-1x=1xx2-1usingimplicit differentiation.

Step by step solution

01

Step 1. Given Information 

We are given sec-1x and using implicit differentiation we need to prove that ddxsec-1x=1xx2-1.

02

Step 2. Proving the expression  

We consider x>1and x<-1,

secsec-1x=x

Differentiating both sides we get,

ddxsecsec-1x=ddxxsecsec-1xtansec-1xddxsec-1x=1ddxsec-1x=1secsec-1xtansec-1xddxsec-1x=1xtansec-1xddxsec-1x=1xx1-1secsec-1x2ddxsec-1x=1xxx2-1xddxsec-1x=1xx2-1

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