Here we look at graphs of higher order derivatives.

Since the derivative gives us a formula for the slope of a tangent line to a curve, we can gain information about a function purely from the sign of the derivative. In particular, we have the following theorem

Below we have graphed :
PIC
Is the first derivative positive or negative on the interval ?
Positive Negative
Below we have graphed :
PIC
Is the graph of increasing or decreasing as increases on the interval ?
Increasing Decreasing

We call the derivative of the derivative the second derivative, the derivative of the derivative of the derivative the third derivative, and so on. We have special notation for higher derivatives, check it out:

First derivative:
.
Second derivative:
.
Third derivative:
.

We use the facts above in our next example.

Here we have unlabeled graphs of , , and :
PIC
Identify each curve above as a graph of , , or .