   Chapter 2.R, Problem 11E

Chapter
Section
Textbook Problem

# 10–11 Find f ' ( x ) from first principles, that is, directly from the definition of a derivative. f ( x ) = x 3 + 5 x + 4

To determine

To find: f'(x)

Explanation

1) Concept:

definition of first derivative

2) Formula:

f'x=limh0fx+h-f(x)h

Identity:  a+b3=a3+3a2b+3ab2+b3

3) Given:

fx=x3+5x+4

4) Calculation:

fx=x3+5x+4

fx+h=(x+h)3+5(x+h)+4

Substitute values of fx and fx+h in f'x=limh0fx+h-f(x)h

We get

f'x=limh0(x+h)3+5x+h+4-(x3+5x+4)h

By using identity a+b3

f'x=limh0x3+3x2h+3xh2+h3+5x+h+4-(x3+5x+4)h

By solving it

f'x=limh0

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