Section 2'3 The Derivative Function

1
Section 2.3The Derivative Function
2

• Fill out the handout with a partner
• So is the table a function?
• Do you notice any other relationships between f
  and its 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?

3
Plot of curve with its derivative
4

• Consider the following graphs
• Which graph is the derivative of the other graph?
  Why?

5
The Derivative Function

• For and function f, the derivative function is
  given by
• For every x-value where this limit exists we say
  that f is differentiable at that x-value.

6
Lets 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

7
Now lets try a couple of power functions

• Use the definition of the derivative to find
  derivatives of the following functions
• It turns out this pattern continues for any
  exponent, say n, of a power function
• We call the following the power rule

8

• Each person will need two sheets of scratch paper
• Draw the graph of a function on one piece of
  paper
• 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)
• Now on a separate sheet of paper, see if you can
  sketch the graph of the derivative
• Pass the graph of the derivative to the person to
  your left
• Now given the graph of the derivative, see if you
  can sketch the graph of the original function
• Compare the sketched original with the original
  graph
• How did you do?