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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.