MCV4U1 – UNIT TWO
UNIT TWO: DERIVATIVES
LESSON ONE: The Derivative Function
The expression for the slope of the tangent line (or the instantaneous rate of change)
lim
h0
value which depends on x.
Therefore, the expression defines a function which we call the DERIVATIVE FUNCTION.
It is given a special notation f x , which we read as “ f prime .“
f x  h  f x 
determines a
h

dy
, y , Dx y, Dx f x . They all mean the same thing.
dx
The derivative of a function f x  at the number a is given by


f a  h  f a
.
f a  lim
h0
h 
When we find a derivative using this formula, we are using the “method” of FIRST PRINCIPLES.


Other notations that are used are.....
A function is said to be differentiable at a point a if f a exists.
Refer to your text on p. 70 for three ways when a derivative doesn’t exist. (CUSP, VERTICAL TANGENT, AND DISCONTINUITY)
Note that a function can be continuous at a point, but not differentiable at that point.
For example, recall: f x   x
 
Examples:
2
1.
Find the derivative f x  if f x   3x  2x .


2.
Find
dy
if
dx

y
x
x 2

3.
Determine an equation of the line that is tangent to the graph of
x  2 y  6  0.
f x   x  2
and parallel to the line