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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

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rule for three
functions and then

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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

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to use that to figure
out the product

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rule for three functions.

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So throughout this
segment, we are

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going to assume that u and 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

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a little easier to write.

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This notation should
be familiar with things

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you saw in the lecture.

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So, for two functions,
let me remind you.

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If u*v, the product, and
you take its derivative,

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so prime 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

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of 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

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to the product of three
functions, u times v times w.

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And we're going to
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

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of what we know to figure out
what this expression will be.

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What this product of
three functions when

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I take its derivative will be.

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

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is 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 essentially takes

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the place of the V up here.

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So using the product
rule for two functions,

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what I get when I
take this derivative,

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is I get u prime times v*w plus,
I take the derivative of this

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second thing,
which is v*w prime.

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And then I leave u alone.

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OK?

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We're not quite done,
but you can see now,

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again if we compare
to what's above,

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you take the derivative
of the first function,

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you leave the second
function alone.

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You take the derivative
of the second function,

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you leave the 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, so let's use it.

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

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that does not look
like a v-- v*w plus,

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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

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in a nice order, so we can see
sort of exactly what happens.

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So, u prime v*w plus v
prime u*w plus w prime u*v.

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So what you can see
here is, what happens?

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You take the derivative
of the first function,

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you leave the second
and third alone.

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Then you take the derivative
of the second function,

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you leave the first
and third alone.

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Then you take the derivative
of the third function,

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you 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 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

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would have four terms
when 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

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tell me what the 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,

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is we're going to 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, and we know
derivatives of the basic trig

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functions, we'll do 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,

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make sure you can do
it, and then you can,

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you can restart the video when
you want to check your answer.

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

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we have an example that
I asked you to determine

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and gave you a moment to do it.

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So now I will actually work
out the example over here

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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, sine x

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is the second, cosine
x is the third.

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So we'll have three terms.

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The first term has to have the
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-- I

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may want to just
write these below.

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OK.

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Now in the next
term, I should take

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

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remember 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--

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oops-- another cosine
x, the third function.

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OK?

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And then the third term,
I take the derivative

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of the third function and I
leave the first and second

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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

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squared times 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

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for someone who was
reading mathematics,

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I would put all of the
polynomials in front

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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

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this and using 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

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of that function, f of x.

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And this is where we'll stop.