
Here we make a connection between a graph of a function and its derivative and higher order derivatives.

Consider the graph of the function $$ below:
On which of the following intervals is $$ increasing?
$$ $$ $$ $$ $$
Which of the following famous functions are increasing on $$?
$$ $$ $$ $$ $$ $$ $$
Since the derivative gives us a formula for the slope of a tangent line to a curve, we can gain information about a function purely from the sign of the derivative. In particular, we have the following theorem
Below we have graphed $$:
Is the function $$ increasing or decreasing on the interval $$?
Increasing Decreasing
We call the derivative of the derivative the second derivative, the derivative of 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.