Performance Considerations

What Is a Derivative?

Differential calculus is used to determine the derivative of a function. A derivative of the function F is another function that describes how the value of F changes with respect to some variable. Most of the examples in this appendix feature functions of x , so most functions are denoted as F(x) . The derivative of a function is denoted as dF(x)/dx . In plain language, this is read as "the change in F(x) with respect to x ". Similarly, you could talk about dy/dx , dz/dx , dz/dy , and many other derivatives.

So, a derivative basically describes the rate of change of the output with respect to the input variable. For a straight line, this rate of change is the slope and it is the same for every point on the line. That isn't the case for other functions. Consider Figure A.3.

Figure A.3: Functions and how they change.

In the leftmost graph, the curve is a straight line. The change in y with respect to x is equal at all points along the curve. In the center graph, the change of y with respect to x is different at all three of the designated points. In fact, it is different for all points along the curve. The rightmost graph shows a sine wave. The value of y is periodic and the rate of change in y is also periodic.

Several chapters demonstrate the value of knowing how variables change with respect to each other. It is not enough to know that they change. You need a way to derive the equation that describes how they change. That is the purpose of the derivative. Before I get into the real math, get out a piece of paper and a pencil and graph out some rough approximations of how the slopes change in each of the graphs in Figure A.3. You should get something that looks like the rough sketches in Figure A.4.

Figure A.4: Graph of the slopes from Figure A.3.

The next sections describe how to find the actual equations for these functions. These derivates are used frequently in this book.