Derivative of a function

Warm-up

lim x-6x ⟶ 6x2-36

1

12∴ limit exists

= -5/0 , ∴ limit DOES NOT exist

lim x2-12x+36x ⟶ 1x2-7x+6

=-18 , ∴ limit existslim x2-81x ⟶ 9 9-x

Definition of the Derivative

EXAMPLE: Find the derivative of f(x) = 3x2 - 2x

f’(x) = [3(x+h)2 - 2(x+h)] - [3x2 - 2x]

hf’(x) = [(3x2 +6xh + 3h2 ) - (2x+2h)] - [3x2 -

2x]h

f’(x) = 3x2 +6xh + 3h2 - 2x - 2h - 3x2 + 2x

hf’(x) = 6xh + 3h2 - 2h

hf’(x) = h (6x + 3h - 2)

h

f’(x) = (6x + 3h - 2)

f’(x) = (6x - 2)

PRACTICE

1. f(x)= 5x- 82. f(x)=-3x +

93. f(x)= 12x + 3x

- 10

5 -3 15

4. f(x) = 4x2 - 5x

5. f(x) = 2x2 - x +5

6. f(x) = -3x2 + 6x - 8

8x - 5 4x - 1 -6x + 6