Chapter 2.1, Problem 69E

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361

Chapter
Section

### Calculus of a Single Variable

11th Edition
Ron Larson + 1 other
ISBN: 9781337275361
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=âˆ’2.

Calculation:

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

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

f'(c)=limxâ†’c+f(x)âˆ’f(c)xâˆ’c=limxâ†’câˆ’f(x)âˆ’f(c)xâˆ’c

Now, find the value of:

limxâ†’c+f(x)âˆ’f(c)xâˆ’c=limxâ†’2+x3+2x2+1âˆ’f(âˆ’2)xâˆ’(âˆ’2)=limhâ†’0(âˆ’2+h)3+2(âˆ’2+h)2+1âˆ’{(âˆ’2)3+2(âˆ’2)2+1}âˆ’2+h+2=limhâ†’0âˆ’8+h3+3Ã—4Ã—hâˆ’3Ã—h2Ã—2+2(4+h2âˆ’4h)+1âˆ’(1)h=limhâ†’0âˆ’8+h3+12hâˆ’6h2+8+2h2âˆ’8hh

Simplify further,

limxâ†’c+f(x)âˆ’f(c)xâˆ’c=limhâ†’0

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