   Chapter 2.1, Problem 69E

Chapter
Section
Textbook Problem

Using the Alternative Form of the Derivative In Exercises 69-76, use the alternative form of the derivative to find the derivative at x = c , if it exists. f ( x ) = x 3 + 2 x 2 + 1 ,   c = − 2

To determine

To calculate: The value of derivative of the function f(x)=x3+2x2+1, at x=c=2, by the use of alternative form of derivative.

Explanation

Given: The function: f(x)=x3+2x2+1, and the value of c=2.

Calculation:

Given function is:f(x)=x3+2x2+1

From the definition of derivative, if the derivative exists then:

f'(c)=limxc+f(x)f(c)xc=limxcf(x)f(c)xc

Now, find the value of:

limxc+f(x)f(c)xc=limx2+x3+2x2+1f(2)x(2)=limh0(2+h)3+2(2+h)2+1{(2)3+2(2)2+1}2+h+2=limh08+h3+3×4×h3×h2×2+2(4+h24h)+1(1)h=limh08+h3+12h6h2+8+2h28hh

Simplify further,

limxc+f(x)f(c)xc=limh0

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