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

f(x)=12-14x,x0;11+ex,x>0

Short Answer

Expert verified

The derivative of function is

f'(x)=-14,x<0-14,x=0-ex1+ex2,x>0

Step by step solution

01

Step 1. Given Information 

The given function isf(x)=12-14x,x0;11+ex,x>0

02

Step 2. Calculation 

From the differentiation rules,

ddx12-14x=-14ddx11+ex=-ex+ex2letg(x)=12-14x,h(x)=11+exAndg'(x)=-14,h'(x)=-ex+ex2

03

Step 3. Calculation

Checkg(0)=h(0)g(0)=12h(0)=12

This is true which follows that the function is continuous at x=0

Checkg'(0)=h'(0)g'(0)=-14h'(0)=-e01+e02=-14

So, although function is continuous at zero, it is always differentiable.

Thus, the derivative of the function is

f'(x)=-14,x<0-14,x=0-ex1+ex2,x>0

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