**Chain Rule**

Suppose that ,
and that each of these variables are in turn functions of .

Then for any variable , we have the following,

*z*is a function of*n*variables,*m*variables,Then for any variable

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 forWe 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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