DERIVATIVE OF A FUNCTION

1.5

DEFINITION OF A DERIVATIVE

• OTHER FORMS:

• OPERATOR:

0

( ) ( )limh

dy f x h f x

dx h

dy

dx'( )f x 'ydf

dx

, , ,

( )df x

dx

DERIVATIVES

• ****THE DERIVATIVE IS A FUNCTION. IF YOU PLUG IN AN X VALUE, THE DERIVATIVE WILL TELL YOU THE SLOPE OF THE TANGENT LINE AT THAT X VALUE.

EXAMPLE 1

• A)FIND THE DERIVATIVE:

• B)FIND THE SLOPE OF THE TANGENT LINE AT X=2

4( )f x x

EXAMPLE 2

• FIND DY/DX: y x

EXAMPLE

• SIMPLIFY:

DIFFERENTIABILITY• A FUNCTION IS DIFFERENTIABLE EVERYWHERE THAT

THE DERIVATIVE EXISTS.

WHAT IT LOOKS LIKE TO NOT BE DIFFERENTIABLE

AT A CORNER

AT A CUSP

AT A VERTICAL TANGENT

AT A DISCONTINUITY

ONE SIDED DERIVATIVES

• SHOW THAT THE FOLLOWING FUNCTION IS NOT DIFFERENTIABLE AT X = 0

2 , 0

2 , 0

x xy

x x

APPROXIMATE THE VALUE OF THE DERIVATIVE

• F(X)

• F’(1)

• F’(3)

• F’(2)

• F’(-1)

SKETCHING THE GRAPH

GIVEN F(X) GRAPH F’(X)

4

2

-2

-4

-5 5 10

HOMEWORK

• PG 144 #25, 26 ANSWER THE QUESTION AND APPROXIMATE THE VALUE OF F’(X1), F’(X7)

• PG 156 #1-31 ODD