Name:
Block:
3.1 Derivative of a Function
Definition of the Derivative
Example 1 Applying the Definition
Differentiate f (x ) = x 3 . ( Find the derivative)
Date: __________ AP Calculus
Alternate Definition of the Derivative at a Point
Example 2 Applying the Alternate Definition
Differentiate f (x ) = x using the alternate definition.
Notation
Calculus Section 3.1 Page2
Example 3 Recognizing a given limit as a derivative
Given f ′( x ) , determine f (x ) .
(x + h)2 − x 2
a) lim
h→ 0
h
3
b) lim
h→ 0
x+h −3 x
h
tan (x + h) − tan x
h→ 0
h
c) lim
x+h
x
− 2
2
( x + h) + 1 x + 1
d) lim
h→ 0
h
e x +h − e x
e) lim
h→ 0
h
1 1
1
f) lim 
− 
h→ 0 h  x + h
x
Example 4 Expressing the derivative as a limit
Express the derivative of each of the following functions as a limit.
a) y = cos x
b) y = 3x
c) y = 3
Calculus Section 3.1 Page3
Relationship between the graphs of
Example 5 Graphing
f′
from
f
and
f′
f
Calculus Section 3.1 Page4
Example 6 Graphing
f
from
f′
Calculus Section 3.1 Page5
One-Sided Derivatives
The Right-hand derivative at a
The Left-hand derivative at a
Example 7 One-Sided Derivatives Can Differ at a Point
Show that the following function has left-hand and right-hand derivatives at x = 0 , but no
derivative there.
x 2 ,
x≤0
f (x ) = 
2x,
x >0
Calculus Section 3.1 Page6