Chain Rule
Statement
The Chain Rule is a theorem of calculus which states that if , then
wherever those expressions make sense.
For example, if ,
, and
, then
.
Here are some more precise statements for the single-variable and multi-variable cases.
Single variable Chain Rule:
Let each of be an open interval, and suppose
and
. Let
such that
. If
,
is differentiable at
, and
is differentiable at
then
is differentiable at
, and
.
Multi-dimensional Chain Rule:
Let and
. (Here each of
,
, and
is a positive integer.) Let
such that
. Let
. If
is differentiable at
, and
is differentiable at
then
is differentiable at
and
. (Here, each of
,
, and
is a matrix.)
Intuitive Explanation
The single-variable Chain Rule is often explained by pointing out that
.
The first term on the right approaches , and the second term on the right approaches
, as
approaches
. This can be made into a rigorous proof. (But we do have to worry about the possibility that
, in which case we would be dividing by
.)
This explanation of the chain rule fails in the multi-dimensional case, because in the multi-dimensional case is a vector, as is
, and we can't divide by a vector.
However, there's another way to look at it.
Suppose a function is differentiable at
, and
is "small". Question: How much does
change when its input changes from
to
? (In other words, what is
?) Answer: approximately
. This is true in the multi-dimensional case as well as in the single-variable case.
Well, suppose that (as above) , and
is "small", and someone asks you how much
changes when its input changes from
to
. That is the same as asking how much
changes when its input changes from
to
, which is the same as asking how much
changes when its input changes from
to
, where
. And what is the answer to this question? The answer is: approximately,
.
But what is ? In other words, how much does
change when its input changes from
to
? Answer: approximately
.
Therefore, the amount that changes when its input changes from
to
is approximately
.
We know that is supposed to be a matrix (or number, in the single-variable case) such that
is a good approximation to
. Thus, it seems that
is a good candidate for being the matrix (or number) that
is supposed to be.
This can be made into a rigorous proof. The standard proof of the multi-dimensional chain rule can be thought of in this way.
Proof
Here's a proof of the multi-variable Chain Rule. It's kind of a "rigorized" version of the intuitive argument given above.
I'll use the following fact. Assume , and
. Then
is differentiable at
if and only if there exists an
by
matrix
such that the "error" function
has the property that
approaches
as
approaches
. (In fact, this can be taken as a definition of the statement "
is differentiable at
.") If such a matrix
exists, then it is unique, and it is called
. Intuitively, the fact that
approaches
as
approaches
just means that
is approximated well by
.
Okay, here's the proof.
Let and
. (Here each of
,
, and
is a positive integer.) Let
such that
. Let
, and suppose that
is differentiable at
and
is differentiable at
.
In the intuitive argument, we said that if is "small", then
, where
. In this proof, we'll fix that statement up and make it rigorous. What we can say is, if
, then
, where
is a function which has the property that
.
Now let's work on . In the intuitive argument, we said that
. In this proof, we'll make that rigorous by saying
, where
has the property that
.
Putting these pieces together, we find that
, where I have taken that messy error term and called it
.
Now, we just need to show that as
, in order to prove that
is differentiable at
and that
.
I believe we've hit a point where intuition no longer guides us. In order to finish off the proof, we just need to look at and play around with it a bit. It's not that bad. For the time being, I'll leave the rest of the proof as an exercise for the reader. (Hint: If
is an
by
matrix, then there exists a number
such that
for all
.)