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CHRISTINE BREINER: Welcome
back to recitation.

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In this segment, we're going to
talk about the product rule

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for three functions and then
we're going to do an example.

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And what I want to do first is
remind you the product rule

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for two functions, because we're
going to use that to

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figure out the product rule
for three functions.

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So throughout this segment, we
are going to assume that U and

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V and W are all functions
of x.

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So I'm going to drop the
of x just so it's a

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little easier to write.

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This notation should be familiar
with things you saw

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in the lecture.

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So, for two functions,
let me remind you.

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If UV, the product, and you take
its derivative, so prime

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will denote d dx.

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Then we can take the derivative
of the first times

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the second function left alone,
plus the derivative of

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the second function times
the first left alone.

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So this should again be
familiar from class.

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And now what we want to do is
expand that to the product of

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three functions, U times V times
W. And we're going to

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explicitly use this rule.

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So, U V W prime is what
we want to look at.

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So we're just going to take
advantage of what we know to

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figure out what this
expression will be.

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What this product of three
functions when I take its

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derivative will be.

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So in order to do this easily,
what we're going to do is

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treat V times W as a
single function.

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OK?

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So V times W will be our
second function that

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essentially takes the place
of the V up here.

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So using the product rule for
two functions, what I get when

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I take this derivative, is I get
U prime times VW plus, I

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take the derivative
of this second

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thing, which is VW prime.

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And then I leave U alone.

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OK?

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We're quite done, but you can
see now, again if we compare

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to what's above, you take the
derivative of the first

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function, you leave the
second function alone.

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You take the derivative of the
second function, you leave the

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first function alone.

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But now again, what
do we do here?

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Well we have a product rule
for two functions,

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so let's use it.

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So, I'll leave the first thing
alone, U prime-- oops, that

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does not look like a UV--

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VW plus, now let's
expand this.

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Take the derivative of the
first function there.

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That's V prime.

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I leave the W alone.

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Plus the derivative of
the second function.

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That's w prime.

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I leave the V alone.

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And I keep the U there.

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OK?

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I'm going to just expand and
write it in a nice order, so

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we can see sort of exactly
what happens.

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So, U prime VW plus V prime UW
plus W prime UV. So what you

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can see here is, what happens?

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You take the derivative of the
first function, you leave the

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second and third alone.

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Then you take the derivative
of the second function, you

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leave the first and
third alone.

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Then you take the derivative
of the third function, you

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leave the first and
second alone.

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And you add up those
three terms.

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I would imagine that
at this point you

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anticipate a pattern.

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So if I had a fourth function.

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If I did U times V times
W times Z, let's say.

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And I took that derivative
with respect to x.

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You could probably anticipate,
you would have four terms when

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you added them up.

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And that fourth term would have
to include a derivative

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of the fourth function.

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So from here, actually you can
probably tell me what the

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derivative of the product
of n functions is.

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And you can check it using
the same kind of rule.

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But what we're going to do at
this point, is we're going to

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just make sure we
understand this.

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We're going to compute
an example.

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So since we know products,
or we know derivatives

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of powers of x.

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And we know derivatives of the
basic trig functions, we'll do

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a product rule using
those functions.

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So, let me take an example.

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So we'll say, f of x equals
x squared sine x cosine x.

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OK?

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And I want you to find
f prime of x.

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Ok, I'm going to give you
a moment to do it.

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You should probably pause the
video here, make sure you can

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do it, and then you can, you can
restart the video when you

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want to check your answer.

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OK, so we have a product role
for three functions, we have

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an example that I asked you to
determine and gave you a

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moment to do it.

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So now I will actually
work out the example

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over here to the right.

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So I will determine
f prime of x.

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Now what are our three
functions?

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Well we have x squared is the
first, sin x is the second,

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cosine x is the third.

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So we'll have three terms. The
first term has to have the

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derivative of the x squared.

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That's going to give me a 2x.

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And I leave the other
two terms alone.

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So I have 2x sine x
cosine x plus--

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I may want to just write
these below.

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OK.

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Now in the next term,
I should take the

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derivative of the sine x.

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And leave the x squared and
the cosine x alone.

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The derivative of sine
x is cosine x.

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So I'm actually going to
write this underneath.

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So we'll have--

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I'm going to put the plus
underneath also so we remember

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it's a sum.

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Plus, so the derivative
of sine x is cosine x.

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And then we have a times x
squared times-- oops--

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another cosine x of the
third function.

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OK?

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And then the third term, I take
the derivative of the

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third function and I leave the
first and second alone.

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The derivative of cosine
x is negative sine x.

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So I actually have a negative
sine x times x squared times

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the sine x here.

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I can do some simplifying
if I want.

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But maybe, if I were trying to
write this nicely for someone

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who was reading mathematics,
I would put all of the

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polynomials in front and all of
the coefficients in front.

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So to be very kind to someone,
I might write it like this.

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And notice cosine x, cosine
x is cosine squared x.

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And then minus x squared
sine squared x.

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And there are other ways, I
could rewrite this and using

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trig identities.

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But this is a sufficient
answer at this point.

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So this is actually a good way
to write the derivative of

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that function f if x.

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And this is where we'll stop.

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