Consider the following simple example. Let f be a function of two variables defined by:
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
But also defines (substituting for u and v in terms of x and y)
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
Consider the function given by . We are used to wring such things as:
iii) . For example we would write for example.
Equally well, of course, it would be true to write .
The meaning of (i) and (ii) are mathematically precise. means the derived function and means the value of at x.
The meaning of can be more devious.
It can simply be taken as synonymous with . That is .
When such is the intention it would be indisputable that means .
But there are other more shady uses as we will see.
Now consider substituting in to define a function F defined as .
The chain rule says
where here means .
Note that F is not equal to f, but mathematicians frequently write the chain rule as,
Here does not mean which is after all .
To see the chain rule in a more precise and unambiguous form think of as defining a function g given by , then and we see the chain rule as saying
Of course the u here is an entirely dummy symbol.