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Chapter 2: Q. 45 (page 222)

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

f(x)={2x,ifx鈮�-2;1x2,ifx>-2}

Short Answer

Expert verified

The derivative of fucntion is

f'(x)=ln2(2x),ifx<-2undefined,ifx=-2-2x-3,ifx>-2

Step by step solution

01

Step 1. Given Information

The given function isf(x)={2x,ifx鈮�-2;1x2,ifx>-2}

02

Step 2. Calculation

From the differentiation rules,

ddx(2x)=ln2路2xddxx-2=-2x-3

Letg(x)=2x,h(x)=1x2Checkg(-2)=h(-2)g(-2)=2-2=14h(-2)=1(-2)2=14

It follows that the function is continuous at x=-2.

Checkg'(-2)=h'(-2)g'(-2)=ln2(2)-2 =ln24h'(-2)=-2(-2)-3 =14

Thus, although function is continuous at x=0, it is not differentiable.

Thus, derivative of function is as follows,

f'(x)=ln2(2x),ifx<-2undefined,ifx=-2-2x-3,ifx>-2

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