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.