Derivatives in Mathematica

There are several ways to find derivatives with Mathematica.  We will only look at a few of these.
Let's find the derivative of a rational function:

The definition can be used to find a derivative, though this method probably takes longer than others.
To use the definition, we first find an expression for the difference quotient.

We next simplify this expression.  Note that you can use % to refer to the last answer given by Mathematica.
Be careful, however, in that it is the order in which you execute things that determines what value % has at
each point, not the order in which things appear on the page.

The derivative is defined as the limit of the difference quotient as h approaches 0.  You can find the
arrow on the BasicInput palette.  (Click on File, Palettes, BasicInput.)  You might also want to use
the AlgebraicManipulation palette as it will show you functions such as Simplify, Expand, Factor,
Together, Apart, etc.

Now let's try a faster method to get the derivative:

Since it is difficult to tell if this matches our previous answer, let's combine the 2 terms of this
last answer:

Now we can see that both methods gave the same result.

Created by Wolfram Mathematica 6.0  (09 January 2008)