WEBVTT

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Functions often come defined
as quotients, so.

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Let's just write that word down
quotients so functions come

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defined. This quotients by,
which we mean we have one

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function cause X, let's say
divided by another function,

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cause X divided by X squared.
What we do is we identify that

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as one function you divided by
another function V.

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This then gives us yet another

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result. Another formula that we
need to be able to remember.

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This one goes VU

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by TX. Minus
you DV by

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DX all over

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V squared. Now it looks very
complicated formula, but it's

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not really you just have to
remember the minus side and I

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must say the way that I always
remember it is if anything is

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going to go wrong with anything,
it's going to be what's in the

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denominator. So when we do the
derivative. We're going to have

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to have a minus sign.

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Let's have a look how this
formula is going to work with

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this particular example.

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So let's have a
look at this example.

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Y equals cause X
over X squared, and

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we've identified this as
being you over VU

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divided by V.

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So you is

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cause X. And
the is X squared. We

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can write down their
derivatives. Do you buy DX

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is minus sign X and
DV by DX is 2

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X? Quote the

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formula. DY by the X

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is. VU
by DX

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minus UDVX
all over

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the squared.

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Again, we're quoting the formula
every time, because that way we

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get to remember it. We get to

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know it. Equals and now we can
plug in the various bits that

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we've got. So V will be X

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squared. Times by DU by
DX. So that's times by minus

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sign X. Minus because of the
minus that there is in the

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formula and then we want you
which is convex.

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Times by and we want DV by
DX, which is 2 X.

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And then all over.

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The squared, and in this case V
is X squared, so that's X

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squared all squared.

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Now again, this doesn't look
very nice. It needs tidying up.

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We need to gather things
together, so if I turn over the

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page, write this expression
again at the top of the page.

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Writing that down
again DYX is

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equal 2.
X squared times

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minus sign X
minus cause X.

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Times 2X all
over X squared

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squared. OK, we need to
simplify this look again for the

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common factor, and there's an X
there in the X. Squared is a

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minus sign there a minus sign
there and an X there. So from

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each term we can take out the
minus sign and the X and on top

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that will leave. As with X Sign

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X. Plus most be a plus
now because we've taken the

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minus sign out plus and here
we've got 2.

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Kohl's X left.

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All over and now X squared all
squared is X to the power 4.

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Having done that, we can see
that there's now a factor of X

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common to both the top, the
numerator and to the bottom the

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denominator. So we can divide
the top and the bottom by X in

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order to simplify it, so the
minus sign stays minus X.

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Sign X +2

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cause X. All
over X cubed and we can leave

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it like that.

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Simplified in that way.

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And that's useful, because now
if we wanted to, we could go on

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and put it equal to 0 and we
could sort out Maxima and minima

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and all that kind of thing. So
it's always helpful to try and

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rearrange these expressions,
particularly to get the top in a

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simplified form. Notice though,
that we didn't cancel any of

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these axes in here. That part of
the sine X. That part of the

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variables. You can't just go
canceling them out. You can only

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cancel those out, which are

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common factors. Let's take a
second example. This time. Let's

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take it one that's just got
polynomial functions of accent,

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so we've got X squared +6
all over 2X.

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Minus 7, just polynomials now
causes no signs etc, so this is

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a U over VA quotient again.

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Let's line this one up. You
equals X squared, and so do

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you find the X will be

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2X. The is 2X
minus Seven, and so DV

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by DX is just two.

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Quote the formula Y

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equals V. You buy

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TX. Minus
UDVX

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All over the squared.

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Now we quoted the formula. We
now in a position to be able to

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substitute in the various pieces
that we need so.

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Why by

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the X?
Is equal 2.

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Now this is VDU by DX
so that's 2X minus 7.

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Times by du by DX,
which is 2 X.

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Minus. U which was
X squared plus six times divided

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by DX, which was just two
and this is all over the

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square so it's all over 2X
minus 7 squared.

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Now again we need to think about
this one. We need to simplify

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it. We need to get together the
various terms and if we look,

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there's a common factor of two
there and there so we can take

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that two out as a common factor
and put it at the front. So we

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have two. Then when we multiply
out with X times by two X, that

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gives us two X squared.

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X times by 7 gives us minus
Seven X. Then we have minus

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this, so it's minus X squared
minus six. Close the bracket.

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All over 2X
minus Seven or

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square. Keep the two outside and
let's simplify the terms inside.

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X squared Minus Seven
X minus 6.

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All over 2X minus Seven
or squared and that's

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now informed. We need to
go on and do something

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else with it we can do.

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3rd example that I'd like to
do with you is one where

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we're going to use this
result in order to help us

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establish something new.

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Now we're going to use this
result to help us prove another

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result. So let's begin with Y
equals 10 X. It's a standard

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function, so we want to be able
to differentiate it in a

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standard way. We want to result.
We can use and just keep on

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using it. So we've got to begin
with the definition of Tan X Tan

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X is defined as being sign X
over cause X, and of course

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that's now a quotient, isn't it?
That's now you over V, because

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we've been. Able to identify the
you over V. Then we can have U

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equals sign X and so do you buy
the X will be cause X.

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We've got the equals cause X
and so DV by the X

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will be minus.

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Sign X. Lots of causes and
signs about, so we need to be

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very, very careful when we do

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the substitution. Close quote
the formula DY by

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the X is VU
by DX minus UDVX

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all over V squared.

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And we're going to make this
substitution, so let's just work

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our way through that.

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The why by DX.

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Will be. Now it's VDU
by the X so V was

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cause X.

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You was sign X, so
its derivative is caused X.

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Minus from the formula.

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You divvy by DX. Now
you was sign X.

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And DV by the X will V was cause
exo DV by the X is minus sign X.

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Oh, over.
The squared and V was cause X,

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so that's all over Cos squared

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X. We now need to simplify the
top so we have cause X times by

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cause X, so that's cost squared
X. We have a minus and minus

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sign, so that's going to give us
a plus and we've sign X times by

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cynex, so we've signed squared

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X. All over cause

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squared X. Equals.

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Now this is a standard result,
well known result cost squared

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plus sign squared is always
equal to 1 cost squared X plus

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sign squared. X is one, so
that's one over cause squared X

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and of course we have another
way of writing one over cost

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squared. One over kozaks we
usually write as being sack X,

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so one over cost squared X. We
would write as SEK squared X.

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And so that's how we
differentiate tab. And now we've

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got a standard result that the
derivative of tangent is sex

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squared X. We can simply quote
that and use it anytime that we

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want to. We take one more
example of using this in the

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same sort of way.

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Let's take the function Y
equals second X.

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Now we know the definition of
Psychics. It's one over cause X.

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One of the ways of doing this
now is to realize that this is a

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quotient. It's AU over V.

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Having identified those, we
can say you equals 1, and so

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do you buy. The X will be
equal to. Now one is just a

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constant, so remember a
constant is about rate of

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derivative is about rate of
change. So if we

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differentiate something
which is constant rate of

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change must be 0.

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The is cause X and
so DV by the X

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will be minus sign X.

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And again, our
formula is DY by

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X is equal to.

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The DU by The

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X. Minus UDV
by the X all

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over V squared.

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So we're going to make the

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substitution now. This is going
to change things slightly if do

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you buy the Axis zero when we
start multiplying by zero sum of

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things may happen so.

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DY by DX is VDU
by DX now remember V

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was cause X times do
you buy DX that was

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zero because you was one.

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Minus U, that's one.

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Times by DV by DX now remember V
was cause X and so its

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derivative is minus sign X and
then all over V squared, which

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is cost squared X. So what we've
got cause X times by zero is

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zero. Anything times by zero is
0 minus. Minus gives us a plus,

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and so we've got sine X over
cause squared X.

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Looks looks like it might be
something else, and remember

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that we've just seen that tan
is sign over cars, so I

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can write this as one over
cause X times sign X over

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cause X here I've got one
over cause is sex X.

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Times sign over cause which is
10 X so I end up with the

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result that the derivative of
sex is sex tinix.

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So. Anytime we want to use the
derivative of sex, we can do so.

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All we do is we just write it
straight down sex tanks and

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that's it. We don't have to work
it all out again.

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And that's the end of Quotients.