Partial Derivatives, Differentials, and Differentiability

Objectives

In this lab you will illustrate the idea of differentiability (that is, `fit-a-bility' with a plane) for a function of two variables. You will also learn how to take derivatives symbolically with Mathematica.

Comparing Differentiable and Non-Differentiable Functions

Define the following two functions in Mathematica:

You may think of both functions as being zero at (0,0). Plot the functions three dimensions using the Plot3D command. Do f and g look differentiable at (0,0)? Why or why not?

Plot a few `slices' of f and g using the Plot command:

       Plot[{f[0,t], f[t,t], f[t,3t]}, {t,-1,1}];

What do these plots tell you about the differentiability of the functions at the point (0,0)?

Symbolic Derivatives

Every derivative Mathematica takes is a partial derivative, consistent with the operational aspect of taking derivates. The command for taking the x-partial of a function f[x,y] is

       fsubx = D[ f[x,y], x]

The part of this command which actually takes the derivative with respect to x is the the D[ , x]. The first part (fsubx = ) is a label that we are giving the derivative to refer to it later. For example, you may wish to Simplify the derivative you just took; since we saved the results in `fsubx' we can say:

       Simplify[fsubx]

or we may wish to put it over a common denominator, in which case we would say

       Together[fsubx]

Plot3D the partials of f and g. That is,

       Plot3D[fsubx, {x,-1,1}, {y,-1,1}, PlotPoints->30];

What does this tell you about the differentiability of these functions at (0,0)?

Finally

Use some of these techniques to check out the differentiability of a homework problem. Try using the symbolic derivative command on some nasty function to see what the derivatives should look like.