Section 2.3
The Derivative Function
• Fill out the handout with a partner
– So is the table a function?
– Do you notice any other relationships between f and it’s
derivative?
• Hint: What is the value of the derivative when f is increasing?
Decreasing?
– What observations did you make about the concavity of the graph?
Plot of curve with its derivative
• Consider the following graphs
• Which graph is the derivative of the other graph? Why?
The Derivative Function
• For and function f, the derivative function is
given by
f ( x  h)  f ( x )
f ' ( x)  lim
h 0
h
• For every x-value where this limit exists we
say that f is differentiable at that x-value.
• Example
– Find the derivative of
f ( x)  x
2
Let’s talk about derivatives of families of functions
• What is the derivative of a constant function (i.e.
f(x) = 5) ?
– It is 0
• What is the derivative of a linear function (i.e.
f(x) = 3x – 4) ?
– It is the slope of the linear function
• These are always true, but we must be able to use
the definition of the derivative in order to prove
these
Now let’s try a couple of power functions
• Use the definition of the derivative to find derivatives
of the following functions
f ( x)  x
g ( x)  x
2
3
• It turns out this pattern continues for any exponent,
say n, of a power function
• We call the following the power rule:
If f ( x)  x then f ' ( x)  nx
n
n 1
1) Each person will need two sheets of scratch paper
2) Draw the graph of a function on one piece of paper
3) Now, give that graph to the person on your right (if
you are all the way to the right, you must give yours
to the person all the way at the left)
4) Now on a separate sheet of paper, see if you can
sketch the graph of the derivative
5) Pass the graph of the derivative to the person to your
left
6) Now given the graph of the derivative, see if you
can sketch the graph of the original function
7) Compare the ‘sketched original’ with the original
graph
8) How did you do?