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Hi.

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Welcome back to recitation.

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In lecture you introduced
the idea of differentials

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and learned how to compute them.

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So I have a couple examples
here for you to do.

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So compute the
differential d of 7

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u to the ninth plus 34 minus
5 u to the minus third.

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And d of sine
theta cosine theta.

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So why don't you take a
minute, work those out

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and we'll come back and
we'll work them out together.

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All right, welcome back.

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So hopefully you had
some luck with these.

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Let's go through them.

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So right, so a
differential is really,

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it's just another
notation for something

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you already know how to do.

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So it's another way of
keeping track of a derivative.

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The thing we don't write is
we don't write the over the d,

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the variable we're
differentiating

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with respect to.

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So we pick that
up from what we're

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taking the differential of.

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So in this case, we
look at this expression.

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Let's do the first one first.

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So we look at d 7 u to the
ninth plus 34 minus 5 u

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to the minus third.

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And we just can distribute
that d through in the same way

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that we can with
ordinary derivatives.

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So OK, so this is
equal to 7 d of u

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to the ninth plus d of 34 minus
5 d of u to the minus third.

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And now we just
do the chain rule.

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So here d of u to the ninth
is 9 u to the eighth du.

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So that du comes out of that
chain rule that we're doing.

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So this is equal to-- so
well, 7 and we drop the 9

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down-- so that's 63 u
to the eighth du plus--

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well OK, d of 34,
34 is a constant.

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It just kills it.

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That's 0.

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Minus 5 d of u to
the minus third.

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So again, u to the minus third.

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That's just a power of u.

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We apply our usual rule for it.

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So it's minus 3 u
to the minus 4 du.

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du from the chain rule.

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Again, and so we have minus
5 times minus 3 is plus 15.

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Good, so I get to
keep my plus sign.

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Plus 15.

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What did I say? u
to the minus 4 du.

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And so that's all
there is to that.

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Now let's do the
second example here.

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We have sine theta cosine theta.

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So same exact idea.

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Here we have a product
rule as our first step.

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So OK, so we take the derivative
of the first times the second.

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So the derivative-- the
differential of the first,

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I should say.

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Right?

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So the product rule
for differentials

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is just the same as the
product rule for derivatives

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except instead of
taking derivatives

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you take differentials.

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So the differential of sine
theta is cosine theta d theta.

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And then times the second.

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Plus the first, sine theta,
times the differential

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of the second, which is
minus sine theta d theta.

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OK.

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And now if we like, we can, you
know, put this all together,

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factor out the d
theta to the end.

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And we can rewrite this
as cosine squared theta

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minus sine squared
theta d theta.

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And of course you could rewrite
this a bunch of other ways

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using your trig identities.

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Just like you could have started
by writing sine theta cosine

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theta as 1/2 sine of 2*theta
before taking the differential.

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All right, so that's really
all there is to that.

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So there you go.

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Differentials.

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I'll end there.