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Hi.

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Welcome back to recitation.

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In lecture introduced the idea
of differentials and learned

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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 7u to the ninth plus 34

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minus 5u 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 and

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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, it's just another

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notation for something 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 taking the

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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.
So we look at d 7u to the

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ninth plus 34 minus 5u
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 7d of
u to the ninth plus d of 34

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minus 5d 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 9u 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 down-- so

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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 5d of u to the
minus third.

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So again, u to the
minus third.

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That's just the power of u.

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We apply our usual
rule for it.

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

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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 4du.

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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

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the second.

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So the derivative, the
differential of the first, I

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should say.

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Right?

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So the product rule for
differentials is just the same

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as the product rule for
derivatives except instead of

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taking derivatives 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 plus
the first sine theta times the

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differential of the second,
which is minus

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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

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together, factor out the
d theta to the end.

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And we can rewrite this as
cosine squared theta minus

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sine squared theta d theta.

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And of course you could rewrite
this a bunch of other

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ways using your trig
identities.

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Just like you could have started
by writing sine theta

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cosine theta as 1/2 sine of
2 theta before taking the

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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.

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