U2S2V12 Recitation video.txt

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Welcome back to recitation.
In this segment, we're going to talk about the product
rule for 3 functions, and then we're going to do an example.
And what I want to do first is remind
you the product rule for 2 functions
because we're going to use that to figure out the product
rule for 3 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 2 functions, let me remind you--
if you have 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 3
functions-- u times v times w.
And we're going to explicitly use this rule.
So uvw 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 3 functions when I take its derivative
will be.
So in order to do this easily, we're
going to do is treat v times w as a single function.
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 2 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 the u alone.
We're not 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 2 functions, so let's use it.
So I'll leave the first thing alone. u prime--
oops, that does not look like a v. 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.
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 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 could check it using this 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 sin x cos x.
And I want you to find f prime of x.

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 restart the video when you want to check your answer.

And coming up in five, four, three--
OK, so we have a product rule for 3 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, cos
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 sin x cos x, plus I may
want to just write these below.
Now in the next term, I should take the derivative of the sin
x, and leave the x squared and the cos x alone.
Derivative of sin x is cos x.
So I'm actually going to write this underneath.
So we'll have-- going to put the plus underneath also
so we remember it's a sum.
Plus-- so the derivative of sin x is cos x,
and then we have a times x squared, times--
oops, another cos x of what--
the third function.

And then the third term, I take the derivative
of the third function, and I leave the first and second
alone.
The derivative of cos x is negative sin x.
So I actually have a negative sin x times x
squared times the sin x here.

I can do some simplifying if I want.
But maybe if I were trying to write this nicely
for someone who is 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, cos x cos x is cos squared x.
And then minus x squared sin 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 of x.
And this is where we'll stop.