Functions often come defined
as quotients, so.

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

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

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

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

as one function you divided by
another function V.

This then gives us yet another

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

This one goes VU

by TX. Minus
you DV by

DX all over

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

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

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

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

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

to have a minus sign.

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

this particular example.

So let's have a
look at this example.

Y equals cause X
over X squared, and

we've identified this as
being you over VU

divided by V.

So you is

cause X. And
the is X squared. We

can write down their
derivatives. Do you buy DX

is minus sign X and
DV by DX is 2

X? Quote the

formula. DY by the X

is. VU
by DX

minus UDVX
all over

the squared.

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

get to remember it. We get to

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

we've got. So V will be X

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

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

formula and then we want you
which is convex.

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

And then all over.

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

squared all squared.

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

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

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

Writing that down
again DYX is

equal 2.
X squared times

minus sign X
minus cause X.

Times 2X all
over X squared

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

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

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

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

that will leave. As with X Sign

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

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

Kohl's X left.

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

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

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

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

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

Sign X +2

cause X. All
over X cubed and we can leave

it like that.

Simplified in that way.

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

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

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

rearrange these expressions,
particularly to get the top in a

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

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

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

cancel those out, which are

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

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

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

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

a U over VA quotient again.

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

you find the X will be

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

by DX is just two.

Quote the formula Y

equals V. You buy

TX. Minus
UDVX

All over the squared.

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

substitute in the various pieces
that we need so.

Why by

the X?
Is equal 2.

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

Times by du by DX,
which is 2 X.

Minus. U which was
X squared plus six times divided

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

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

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

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

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

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

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

gives us two X squared.

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

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

All over 2X
minus Seven or

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

X squared Minus Seven
X minus 6.

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

now informed. We need to
go on and do something

else with it we can do.

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

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

establish something new.

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

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

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

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

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

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

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

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

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

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

will be minus.

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

very, very careful when we do

the substitution. Close quote
the formula DY by

the X is VU
by DX minus UDVX

all over V squared.

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

our way through that.

The why by DX.

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

cause X.

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

Minus from the formula.

You divvy by DX. Now
you was sign X.

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

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

so that's all over Cos squared

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

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

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

cynex, so we've signed squared

X. All over cause

squared X. Equals.

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

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

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

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

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

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

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

got a standard result that the
derivative of tangent is sex

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

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

same sort of way.

Let's take the function Y
equals second X.

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

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

quotient. It's AU over V.

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

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

constant, so remember a
constant is about rate of

derivative is about rate of
change. So if we

differentiate something
which is constant rate of

change must be 0.

The is cause X and
so DV by the X

will be minus sign X.

And again, our
formula is DY by

X is equal to.

The DU by The

X. Minus UDV
by the X all

over V squared.

So we're going to make the

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

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

things may happen so.

DY by DX is VDU
by DX now remember V

was cause X times do
you buy DX that was

zero because you was one.

Minus U, that's one.

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

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

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

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

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

Looks looks like it might be
something else, and remember

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

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

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

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

result that the derivative of
sex is sex tinix.

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

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

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

And that's the end of Quotients.