Consider the following simple example. Let f be a function of two variables defined by:

where .

By substituting,

Let us define a function F of two variables by

Then we can calculate

With the obvious intentions regarding the partial derivatives we can also calculate

It can also be checked that

These two results are indeed true for a general function and substitution . They are regarded as a generalisation of the chain rule for one variable. Again and are often confusingly written as and .

The rule is then

It is very important to remember what is meant by all these items.

Note the chain rule can be used both ways.

ie let , , define

We have

But also defines (substituting for u and v in terms of x and y)

So,

and, of course, we put F as f throughout.

These results are special cases of the so called general chain rule.

If we have a function where x are substitutions

Then

###### Special cases

i)

ii)