Friday, June 29, 2012

Chain Rule

Chain Rule

Suppose that z is a function of n variables, , and that each of these variables are in turn functions of m variables, .
 Then for any variable ,  we have the following,

Wow.  That’s a lot to remember.  There is actually an easier way to construct all the chain rules that we’ve discussed in the section or will look at in later examples.  We can build up a tree diagram that will give us the chain rule for any situation.  To see how these work let’s go back and take a look at the chain rule for 
  given that , , .
 We already know what this is, but it may help to illustrate the tree diagram if we already know the answer.  For reference here is the chain rule for this case,

Here is the tree diagram for this case.


We start at the top with the function itself and the branch out from that point.  The first set of branches is for the variables in the function.  From each of these endpoints we put down a further set of branches that gives the variables that both x and y are a function of.  We connect each letter with a line and each line represents a partial derivative as shown.  Note that the letter in the numerator of the partial derivative is the upper “node” of the tree and the letter in the denominator of the partial derivative is the lower “node” of the tree.

To use this to get the chain rule we start at the bottom and for each branch that ends with the variable we want to take the derivative with respect to (s in this case) we move up the tree until we hit the top multiplying the derivatives that we see along that set of branches.  Once we’ve done this for each branch that ends at s, we then add the results up to get the chain rule for that given situation.

Note that we don’t usually put the derivatives in the tree.  They are always an assumed part of the tree.

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