CHRISTINE BREINER: Welcome back to recitation.
In this segment, we're going to talk about the product rule
for three functions and then we're going to do an example.
And what I want to do first is remind you the product rule
for two functions, because we're going to use that to
figure out the product rule for three functions.
So throughout this segment, we are going to assume that U and
V and W are all functions of x.
So I'm going to drop the of x just so it's a
little easier to write.
This notation should be familiar with things you saw
in the lecture.
So, for two functions, let me remind you.
If UV, the product, and you take its derivative, so prime
will denote d dx.
Then we can take the derivative of the first times
the second function left alone, plus the derivative of
the second function times the first left alone.
So this should again be familiar from class.
And now what we want to do is expand that to the product of
three functions, U times V times W. And we're going to
explicitly use this rule.
So, U V W prime is what we want to look at.
So we're just going to take advantage of what we know to
figure out what this expression will be.
What this product of three functions when I take its
derivative will be.
So in order to do this easily, what we're going to do is
treat V times W as a single function.
OK?
So V times W will be our second function that
essentially takes the place of the V up here.
So using the product rule for two functions, what I get when
I take this derivative, is I get U prime times VW plus, I
take the derivative of this second
thing, which is VW prime.
And then I leave U alone.
OK?
We're quite done, but you can see now, again if we compare
to what's above, you take the derivative of the first
function, you leave the second function alone.
You take the derivative of the second function, you leave the
first function alone.
But now again, what do we do here?
Well we have a product rule for two functions,
so let's use it.
So, I'll leave the first thing alone, U prime-- oops, that
does not look like a UV--
VW plus, now let's expand this.
Take the derivative of the first function there.
That's V prime.
I leave the W alone.
Plus the derivative of the second function.
That's w prime.
I leave the V alone.
And I keep the U there.
OK?
I'm going to just expand and write it in a nice order, so
we can see sort of exactly what happens.
So, U prime VW plus V prime UW plus W prime UV. So what you
can see here is, what happens?
You take the derivative of the first function, you leave the
second and third alone.
Then you take the derivative of the second function, you
leave the first and third alone.
Then you take the derivative of the third function, you
leave the first and second alone.
And you add up those three terms.
I would imagine that at this point you
anticipate a pattern.
So if I had a fourth function.
If I did U times V times W times Z, let's say.
And I took that derivative with respect to x.
You could probably anticipate, you would have four terms when
you added them up.
And that fourth term would have to include a derivative
of the fourth function.
So from here, actually you can probably tell me what the
derivative of the product of n functions is.
And you can check it using the same kind of rule.
But what we're going to do at this point, is we're going to
just make sure we understand this.
We're going to compute an example.
So since we know products, or we know derivatives
of powers of x.
And we know derivatives of the basic trig functions, we'll do
a product rule using those functions.
So, let me take an example.
So we'll say, f of x equals x squared sine x cosine x.
OK?
And I want you to find f prime of x.
Ok, I'm going to give you a moment to do it.
You should probably pause the video here, make sure you can
do it, and then you can, you can restart the video when you
want to check your answer.
OK, so we have a product role for three functions, we have
an example that I asked you to determine and gave you a
moment to do it.
So now I will actually work out the example
over here to the right.
So I will determine f prime of x.
Now what are our three functions?
Well we have x squared is the first, sin x is the second,
cosine x is the third.
So we'll have three terms. The first term has to have the
derivative of the x squared.
That's going to give me a 2x.
And I leave the other two terms alone.
So I have 2x sine x cosine x plus--
I may want to just write these below.
OK.
Now in the next term, I should take the
derivative of the sine x.
And leave the x squared and the cosine x alone.
The derivative of sine x is cosine x.
So I'm actually going to write this underneath.
So we'll have--
I'm going to put the plus underneath also so we remember
it's a sum.
Plus, so the derivative of sine x is cosine x.
And then we have a times x squared times-- oops--
another cosine x of the third function.
OK?
And then the third term, I take the derivative of the
third function and I leave the first and second alone.
The derivative of cosine x is negative sine x.
So I actually have a negative sine x times x squared times
the sine x here.
I can do some simplifying if I want.
But maybe, if I were trying to write this nicely for someone
who was reading mathematics, I would put all of the
polynomials in front and all of the coefficients in front.
So to be very kind to someone, I might write it like this.
And notice cosine x, cosine x is cosine squared x.
And then minus x squared sine squared x.
And there are other ways, I could rewrite this and using
trig identities.
But this is a sufficient answer at this point.
So this is actually a good way to write the derivative of
that function f if x.
And this is where we'll stop.