U2S4V08 Using the Chain Rule.txt

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We're looking at this function p of x.
It's a composition of the cosine function and the cubing
function-- but in what order?
Which one is the inner function and which
is the outer function?
Well, in order to figure that out,
we should think about what we would
do in order to calculate p.
If we want p of x, we'd first take the cosine of x,
and then we cube the result. So the cubing
is being applied to the cosine.
That tells us that the cubing is the outer function.
And the cosine is the inner function.
OK, great.
Now, if we want to take the derivative of this composition,
we need the chain rule.
So here it is to remind you.
If h is f of g of x-- so here f is the outer function,
and g is the inner function-- then, for h prime,
we need to take f prime, the derivative
of the outer function, applied to the inner function.
And then we multiply by g prime, which
is the derivative of the inner function.
Sometimes what people do to make the notation a little easier
is to introduce a new variable to represent
the inner function.
And this is exactly the intermediate variable u
that we used in the last video.
So I'll write it up here.
u is equal to g of x, the inner function.
And so h of x equals f of g of x.
And that's now f of u.
And our derivative h prime of x becomes f prime of u times
g prime of x.

So for our function here, p of x is u cubed,
where u is this inner function cosine x.
And the first part of the chain rule
tells us to take the derivative of this outer function applied
to u.
So we just differentiate the cubing function as normal,
as if u was our original variable.
So we're going to get 3u squared.
But-- but, but, but, but, but, we're not done.
The chain rule says that we also need the derivative
of this inner function.
So we have to have a u prime tagging along.
In Leibniz notation, the u prime is du dx.
And if we set y to be equal to this p of x,
then the 3u squared is dy du.
And the overall derivative that we're going for is dy dx.
And that's our chain rule in the Leibniz form.
OK, well, now we can put everything
in terms of our original variable x.
So, 3u squared is just 3 times cosine x squared.
And the derivative of u, well, that's
the derivative of cosine x.
And that's going to be minus sine x.
And so that's our answer.
We're done.

Let's do another quick example.
So let's say that this time, instead of cosine cubed x,
we have cosine of x cubed.
So you might look at this function
and say, gosh, if only this were cosine of x,
it would be really, really easy.
But it's not.
It's cosine of some stuff.
And whenever you see that, that's
an indication that you might replace the stuff with u.
And so when we do that, this becomes cosine of u,
where u equals x cubed.
And writing it like this tells us
exactly how the chain rule is going to play out.
We have an outer function of u.
And, generally, you want the outer function
to be something pretty basic.
And then we have our inner function, which is u itself.
So the chain rule tells us to take
the derivative of the outer function applied
to u-- so in this case, the derivative of cosine
is minus sign, and we're applying that to u--
and then we need the derivative of u, which is 3x squared.
So putting everything back in terms of x,
we end up with minus sine of x cubed times 3x squared.
And that is our derivative.

So something I want to point out, both in this derivation
and in the previous one, we took into account
both the derivative of the cubing
function and the derivative of the cosine function.
In general, if you have some complicated function
that you want to differentiate, every piece of it
will need to be differentiated at some point in the process.
So if you look back at your original function--
and maybe it has some junk floating around here
but there's a tangent in the middle of it--
and you say to yourself, hey, I don't
remember thinking about the derivative of tangent
at any point in the process, then
unless it's just the tangent of a constant,
you've probably messed up the chain rule
somewhere along the way.
Of course, everyone messes up the chain rule
when they first start using it.
But at least now you know a way of catching some mistakes.
Why don't you go play around with some examples on your own?