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Chapter 2: Q. 25 (page 184)

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

25.f(x)=1x,x=-1

Short Answer

Expert verified

f'(-1)=-1.

Step by step solution

01

Part (a) Step 1. Given information.

A function is given asf(x)=1xandx=c=-1.

02

Part (a) Step 2. Find f'(-1) using h→0 definition of the derivative.

We have

f'(c)=limh鈫�0f(c+h)-f(c)hf'(-1)=limh鈫�0f(-1+h)-f(-1)h=limh鈫�0(1h-1)-(1-1)h=limh鈫�01h-1+1h=limh鈫�01+h-1h(h-1)=limh鈫�0hh(h-1)=limh鈫�01h-1=10-1=1-1=-1

03

Part (b) Step 1. Find f'(-1) using z→-1 definition of the derivative.

We have

f'(c)=limz鈫�cf(z)-f(c)z-cf'(-1)=limz鈫�-1f(z)-f(-1)z-(-1)=limz鈫�-11z-(1-1)z+1=limz鈫�-11z+1z+1=limz鈫�-11+zz(z+1)=limz鈫�-11z=1-1=-1

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