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Describe the derivatives of each of the piecewise-defined functions in Exercises 45–48.

f(x)={ln(-x),ifx<0;lnx,ifx0}

Short Answer

Expert verified

The derivative of function is

f(x)=1x,x<0DNE,x=01x,x>0

Step by step solution

01

Step 1. Given Information 

The given function is f(x)={ln(-x),ifx<0;lnx,ifx0}

02

Step 2. Calculation 

From the differentiation rules,

ddx(ln(-x))=1xddx(lnx)=1xLetg(x)=ln(-x),h(x)=ln(x)Andg'(x)=1x,h'(x)=1xWhenx=0then,g'(0)=h'(0)=undefined

It follows that the function is not continuous at x=0.

Thus, the derivative of function is,

f(x)=1x,x<0DNE,x=01x,x>0

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