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

We say that a function is increasing on an interval if , for all pairs of numbers , in such that .

We say that a function is decreasing on an interval if , for all pairs of numbers , in such that .

We say that a function is decreasing on an interval if , for all pairs of numbers , in such that .

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

A function is increasing on any interval where , for all in .

A function is decreasing on any interval where , for all in .

A function is decreasing on any interval where , for all in .

Below we have graphed :

Is the function increasing or decreasing on the interval ?

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:

Increasing Decreasing

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

We use the facts above in our next example.

Here we have unlabeled graphs of , , and :

Identify each curve above as a graph of , , or .

Here we see three curves, , , and . Since is positive negative increasing decreasing when is positive and positivenegativeincreasingdecreasing when is negative, we see Since is increasing when is positivenegativeincreasing decreasing and decreasing when is positivenegativeincreasingdecreasing , we see Hence , , and .

Here we have unlabeled graphs of , , and :

Identify each curve above as a graph of , , or .

Here we see three curves, , , and . Since is positivenegativeincreasingdecreasing when is positive and positivenegativeincreasingdecreasing when is negative, we see Since is increasing when is positivenegativeincreasingdecreasing and decreasing when is positivenegativeincreasingdecreasing , we see Hence , , and .

Here we have unlabeled graphs of , , and :

Identify each curve above as a graph of , , or .

Here we see three curves, , , and . Since is positivenegativeincreasingdecreasing when is positive and positivenegativeincreasingdecreasing when is negative, we see Since is increasing when is positivenegativeincreasingdecreasing and decreasing when is positivenegativeincreasingdecreasing , we see Hence , , and .