We’ll start off the proof by defining and noticing that in terms of this
definition what we’re being asked to prove is,
Let’s take a look at the derivative of (again, remember we’ve defined and so u
really is a function of x) which we
know exists because we are assuming that is differentiable. By definition we have,
Note as well that,
Now, define,
and notice that
and so is continuous at
Now if we
assume that we can rewrite the definition of to get,
(1)
Now, notice that (1)
is in fact valid even if we let and so is valid for any value of h.
Next, since we
also know that is differentiable we can do something
similar. However, we’re going to use a
different set of letters/variables here for reasons that will be apparent in
a bit. So, define,
we can go
through a similar argument that we did above so show that is continuous at and that,
(2)
Do not get
excited about the different letters here all we did was use k instead of h and let . Nothing fancy here, but the change of
letters will be useful down the road.
Okay, to this
point it doesn’t look like we’ve really done anything that gets us even close
to proving the chain rule. The work
above will turn out to be very important in our proof however so let’s get
going on the proof.
What we need to
do here is use the definition of the derivative and evaluate the following
limit.
(3)
Note that even
though the notation is more than a little messy if we use instead of u we need to remind ourselves here that u really is a function of x.
Let’s now use (1)
to rewrite the and yes the notation is going to be
unpleasant but we’re going to have to deal with it. By using (1),
the numerator in the limit above becomes,
If we then define and we can use (2)
to further write this as,
Notice that we were able to cancel a to simplify things up a little. Also, note that the was intentionally left that way to keep the
mess to a minimum here, just remember that here as that will be important here in a
bit. Let’s now go back and remember
that all this was the numerator of our limit, (3). Plugging this into (3)
gives,
Notice that the
h’s canceled out. Next, recall that and so,
But, if ,
as we’ve defined k anyway, then by
the definition of w and the fact
that we know is continuous at we also know that,
Also, recall
that . Using all of these facts our limit becomes,
This is exactly
what we needed to prove and so we’re done.
