   Chapter 2.8, Problem 28E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f ( x ) = x 2 − 1 2 x − 3

To determine

To find: The derivative of the function f(x)=x212x3 and state the domain of the function and its derivative.

Explanation

Formula used:

The derivative of a function f, denoted by f(x) is,

f(x)=limh0f(x+h)f(x)h (1)

Calculation:

Obtain the derivative of the function f(x)

Use the equation (1) to compute f(x).

f(x)=limh0f(x+h)f(x)h=limh0((x+h)212(x+h)3)(x212x3)h=limh0((x+h)21)(2x3)(x21)(2(x+h)3)(2(x+h)3)(2x3)h=limh0(x2+h2+2xh1)(2x3)(x21)(2x+2h3)h(2x+2h3)(2x3)

Expand the numerator and simplifying further as follows,

f(x)=limh0((2x(x2+h2+2xh1)3(x2+h2+2xh1)x2(2x+2h3)+1(2x+2h3))h(2x+2h3)(2x3))=limh0((2x3+2xh2+4x2h2x3x23h26xh+32x32x2h+3x2+2x+2h

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