derivative of a function 1.5. definition of a derivative other forms: operator:,,,
TRANSCRIPT
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