   Chapter 2.8, Problem 27E ### 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. g ( x ) = 9 − x

To determine

To find: The derivative of the function g(x)=9x 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)

Difference of squares formula: (a2b2)=(a+b)(ab)

Calculation:

Obtain the derivative of the function g(x).

Use the equation (1) to compute g(x),

g(x)=limh0f(x+h)f(x)h=limh0(9(x+h))(9x)h

Multiply both the numerator and the denominator by the conjugate of the numerator,

g(x)=limh0(9(x+h))(9x)h×(9(x+h))+(9x)(9(x+h))+(9x)=limh0((9(x+h))(9x))((9(x+h))+(9x))h((9(x+h))+(9x))

Apply the difference of square formula,

g(x)=limh0(9(x+h))2(9x)2h((9(x+h

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