THE DERIVATIVE FUNCTION
(from First Principles)
Example 
Determine an equation to represent the slope of the tangent at any
point along the function, f(x) = x2.
mtgt  lim
h0
f (a  h)  f (a)
h
What is the slope of the tangent at x = 1? At x = 0? At x = –2?
Note: When f(x) is a quadratic function, f ’(x) is a linear function!
The equation of the slope of the tangent is called the derivative of the function.
THE DERIVATIVE FUNCTION:
The derivative of a function f(x), denoted as f '( x ) , is defined by:
f ' ( x)  lim
h 0
f ( x  h)  f ( x )
h
(provided the limit exists)
Note: When the above limit is used to determine the derivative of a function,
it is called “determining the derivative using first principles”.
The given limit has the following interpretations:
lim
h0
f (a  h)  f (a)
h
 slope of the tangent to y = f(x) at the point where x = a
 IRC of y = f(x) with respect to x when x = a
 the derivative of y = f(x) with respect to x at x = a
DERIVATIVE NOTATION
If y = f(x), then the derivative can be written as:
Lagrange Notation:
Leibniz Notation:
The differentiation operator :
f '( x ) or y '
dy
dx
this is NOT a fraction!!
d 2
( x )  2x
dx
(Note: “derivative” is the noun and “differentiate” is the verb!!)
Example 
Determine the derivative of f ( x )  x  2 using first principles.
Example 
Determine the derivative of f ( x ) 
x 1
using first principles.
3x  2
A function f(x) is differentiable at x = a if f '(a) exists. Four common ways where a
derivative fails to exist are shown below:
A.
A CUSP
B.
y
A CORNER
y
x
C.
A VERTICAL TANGENT
y
x
D.
A DISCONTINUITY
y
x
x
Note: It is possible for a function to be continuous at a point and not
differentiable at that point.
Example 
A.
State the domain on which each function is differentiable:
B.
C.
Homework: p.73–75 #1–3, 6abd, 7–9, 13, 15