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Chapter 2: Q. 47 (page 222)

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

f(x)={x2,x<1;1+lnx,x鈮�1}

Short Answer

Expert verified

The derivative of function is


f'(x)=2x,x<1DNE,x=11x,x>1

Step by step solution

01

Step 1. Given Information 

The given function is f(x)={x2,x<1;1+lnx,x鈮�1}

02

Step 2. Calculation 

From the differentiation rules,

ddx(x2)=2xddx(1+lnx)=1xLetg(x)=x2,h(x)=1+lnxAndg'(x)=2x,h'(x)=1xCheckg'(1)=h'(1)g'(1)=2(1)=2h'(1)=11=1

It follows that the function is not continuous at x=1.

Thus, the derivative of function is,

f'(x)=2x,x<1DNE,x=11x,x>1

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