Summary

Since the derivative of a function f is itself a function, we can consider taking its derivative. The derivative of the derivative is called the second derivative. Similarly, we can consider third derivatives, fourth derivatives, etc. Like the derivative, higher derivatives have special significance to the shape of the graph of a function. We'll see precisely how in this section.

Notation for higher derivatives: The second derivative of y=f(x) can be written using any of the following notations:

The third derivative can be written in a similar way:

For , the derivative is written . The parentheses indicate that this is not a power, but rather a derivative.

Example:

Everyone's favorite example to illustrate the second derivative is motion: for example, the position of a car, its velocity and the acceleration represent f, , and respectively.

Example:

Just as the values of x such that f'(x)=0 are significant in the graph of f, so are the values of x such that are significant: but rather than indicating points at which the extrema may occur, values of x such that indicate points at which changes in concavity may occur (concavity is the umbrella-ness or bowl-ness of a graph).

Concavity:

• A function is concave up on an interval (a,b) if the graph of the function lies above its tangent line on that interval.
• A function is concave down on an interval (a,b) if the graph of the function lies below its tangent line on that interval.
• A point at which the graph changes concavity is called an inflection point. At a point of inflection, the second derivative is either 0 or it doesn't exist.

Second Derivative Test for relative extrema: Let exist on some open interval containing c, and let f'(c)=0.

1. If , then f(c) is a relative minimum;
2. If , then f(c) is a relative maximum;
3. If , then the test gives no information.

An example of an application of the point of inflection is the point of diminishing returns in economics.