Aaron BarkerDEFINITION OF THE

DERIVATIVE

The Definition of the Derivative can be used to find the derivative of polynomials and exponential functions.

To find the derivative of polynomials and exponential functions, use the equation:

THE DEFINITION OF THE DERIVATIVE

First, start by plugging your f(x) into the equation:

Next, simplify the numerator and the denominator as much as possible.

Lastly, substitute 0 for h, therefore all terms containing h will equal 0.

RULES/STEPS TO SOLVING

Find the derivative of f(x) = x3 + 1

I. f’(x) = lim (x+h)3 + 1 – (x3 + 1) h->0 h

II. f’(x) = lim x3 + 3x2h + 3xh2 + h3 +1– x3 – 1 h->0 h III. f’(x) = lim 3x2h + 3xh2 + h3

h->0 h IV. f’(x) = lim 3x2 + 3xh + h2

h ->0

V. f’(x) = 3x2

EXAMPLE #1

Find the derivative of f(x) = 3x2 + x

I. f’(x) = lim 3(x + h)2 + (x+h) – (3x2 + x) h->0 h II. f’(x) = lim 3(x2 + 2xh + h2) + x + h – 3x2 – x h->0 h III. f’(x) = lim 3x2 + 6xh + 3h2 + x + h – 3x2 – x h->0 h IV. f’(x) = lim 6xh + 3h2 + h h->0 h

EXAMPLE #2

V. f’(x) = lim 6x + 3h + 1 h->0

VI. f’(x) = 6x + 1

EXAMPLE #2 CONTINUED

Find the derivative of f(x) = 9x2 + 5x.

PRACTICE PROBLEM #1

Find the derivative of f(x) = 9x2 + 5x.

f’(x) = 18x + 5

PRACTICE PROBLEM #1 SOLUTION

Find the derivative of f(x) = 2x3 + 12x - 18

PRACTICE PROBLEM #2

Find the derivative of f(x) = 2x3 + 12x - 18

f’(x) = 6x2 + 12

PRACTICE PROBLEM #2 SOLUTION