Chain Rule for Partial Derivatives

Objectives

In this lab we will get more comfortable using some of the symbolic power of Mathematica. In the process we will explore the Chain Rule applied to functions of many variables.

Derivatives Along Paths

A function is a rule that assigns a single value to every point in space, e.g. w=f(x,y) assigns the value w to each point (x,y) in two dimensional space. If we define a parametric path x=g(t), y=h(t), then the function w(t) = f(g(t),h(t)) is univariate along the path. The derivative can be found by either substitution and differentiation,

or by the Chain Rule,

Let's pick a reasonably grotesque function,

First, define the function for later usage:

       f[x_,y_] := Cos[
                        x^2 y - Log[ (y^2 +2)/(x^2+1) ]	
                      ]

Now, let's find the derivative of f along the elliptical path , . First, by direct substitution. Find w(t);

       woft = f[ 4 Cos[t], 9 Sin[t] ]

The derivative can then be found using

       dwdt1 = D[ woft, t]

How would you have liked to do that by hand?

Now let's try using the Chain Rule. First, define the path variables:

       xx = 4 Cos[t]; yy = 9 Sin[t];

and now let's do the Chain Rule:

       crule = D[f[x,y],x] D[xx,t] + D[f[x,y],y] D[yy,t]

Notice that this has variables x,y and t. That is because we have not substituted the path in for x and y. To do this, we will use the substitution operation in Mathematica, `/. ->'. Try this to find the final form of :

       dwdt2 = crule/.{x->4 Cos[t], y->9 Sin[t]}

To see that the two methods yield the same answer, try subtracting them and simplifying:

       Simplify[dwdt1 - dwdt2]

If you did everything correctly, the result should be `0.'

Multi-Variable Chain Rule

Essentially the same procedures work for the multi-variate version of the Chain Rule. Try finding and where r and are polar coordinates, that is and . First, take derivatives after direct substitution for ,

       wrtheta = f[ r Cos[theta], r Sin[theta] ]

Then try using the Chain Rule directly,

and then substituting, which in Mathematica can be accomplished using the substitution

       /.{x -> r Cos[theta], y -> r Sin[theta]}

Finally

Try a couple of homework problems. In particular, you may want to give some of the implicit differentiation problems a whirl.