1. Aaron Barker DEFINITION OF THE DERIVATIVE

2.  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

3.  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

4.  Find the derivative of f(x) = x 3 + 1  I. f’(x) = lim (x+h) 3 + 1 – (x 3 + 1) h->0 h  II. f’(x) = lim x 3 + 3x 2 h + 3xh 2 + h 3 +1– x 3 – 1 h->0 h  III. f’(x) = lim 3x 2 h + 3xh 2 + h 3 h->0 h  IV. f’(x) = lim 3x 2 + 3xh + h 2 h->0  V. f’(x) = 3x 2 EXAMPLE #1

5.  Find the derivative of f(x) = 3x 2 + x  I. f’(x) = lim 3(x + h) 2 + (x+h) – (3x 2 + x) h->0 h  II. f’(x) = lim 3(x 2 + 2xh + h 2 ) + x + h – 3x 2 – x h->0 h  III. f’(x) = lim 3x 2 + 6xh + 3h 2 + x + h – 3x 2 – x h->0 h  IV. f’(x) = lim 6xh + 3h 2 + h h->0 h EXAMPLE #2

6.  V. f’(x) = lim 6x + 3h + 1 h->0  VI. f’(x) = 6x + 1 EXAMPLE #2 CONTINUED

7.  Find the derivative of f(x) = 9x 2 + 5x. PRACTICE PROBLEM #1

8.  Find the derivative of f(x) = 9x 2 + 5x.  f’(x) = 18x + 5 PRACTICE PROBLEM #1 SOLUTION

9.  Find the derivative of f(x) = 2x 3 + 12x - 18 PRACTICE PROBLEM #2

10.  Find the derivative of f(x) = 2x 3 + 12x - 18  f’(x) = 6x 2 + 12 PRACTICE PROBLEM #2 SOLUTION