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Chapter 2: Q. 23 (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.

23.f(x)=x2,x=-3

Short Answer

Expert verified

f'(-3)=-6.

Step by step solution

01

Part (a) Step 1. Given information.

A function is given asf(x)=x2andx=c=-3.

02

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

We have

f'(-3)=limh鈫�0f(-3+h)-f(-3)h=limh鈫�0(-3+h)2-(-3)2h=limh鈫�09-6h+h2-9h=limh鈫�0-6h+h2h=limh鈫�0h(h-6)h=limh鈫�0(h-6)=0-6=-6

03

Part (b) Step 2. Find f'(c) using z→(-3) definition of the derivative.

We have

f'(c)=limz鈫�cf(z)-f(c)z-cf'(-3)=limz鈫�-3z2-(-3)2z-(-3)=limz鈫�-3z2-9z+3=limz鈫�-3(z-3)(z+3)(z+3)=limz鈫�-3(z-3)=[(-3)-3]=-6

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