To use the Derivatives utility, choose the command "Derivatives" from the Utilities Menu. The menu bar will disappear, and the Derivative utility will fill the screen. This utility can find the first and second derivatives of a function and display the results either as graphs or as formulas.
A formula to be differentiated can use constants, the variable x, the arithmetic operators +, -, *, /, and ^, and the built-in functions sin, cos, tan, cot, sec, csc, arctan, arcsin, exp, ln, sqrt, abs, and trunc. In addition, it can contain references to other functions that may have been defined and added to the List of Functions window. This feature allows you to graph the derivatives of functions input as graphs or as tables, as well as those input as mathematical formulas. However, when you display the derivatives as formulas, the derivative of a user-defined function such as f(x) will appear simply as f'(x), and the second derivative will appear as f''(x).
After entering the function and resetting xmin, xmax, ymin, and ymax if necessary, just click the "Take Derivatives" button. Either graphs or formulas will be displayed, depending on the setting of a pair of radio buttons located just above the display area. (You can change from one display to another by clicking on the appropriate radio button.) It may take some time to do the calculations before anything is displayed.
If you have chosen the graph display, the display area will be divided into three rectangles containing the graphs of the original function, its derivative, and its second derivative. If you move the mouse over the first of these rectangles, a tangent line to the graph will be displayed. Only the x-coordinate of the mouse location is important--the tangent line at that x-value will be drawn. At the same time, the corresponding point on the graph of the first derivative will be marked with a cross cursor. (The y-coordinate of this point is the slope of the tangent line.) As you move the mouse, the tangent line will sweep through various positions on the curve. If you position the mouse over the graph of the first derivative, a tangent line will be drawn on that graph, with the corresponding point marked on the graph of the second derivative. More stuff is drawn if you move the mouse over the graph of the second derivative.
If you choose the formula display, you will see not just the formulas for the
derivatives, but also steps in the calculation of the derivative. A "D" is used to represent the derivative operation, so that D[sin(x)^2] represents the derivative of sin(x)^2. On the next line, you would see that this derivative is equal to sin(x)*2*D[sin(x)], which will be converted, finally, to sin(x)*2*cos(x). On a color monitor, the derivatives that still need to be computed will be printed in red or blue--the one in red is the one that will be expanded next. For complicated functions, you might find the formulas are not very useful, since they can get very hard to read. In fact, for very complicated functions, the second derivative will be printed just once, without all the intermediate steps. Unfortunately, this program is not capable of doing even elementary simplifications of the formulas. (This is why the calculations for the graphs can take so long.)
