The graph of Cynex looks like

this. And we can see the
gradient of the tangents at

certain points of the graph.

For instance, when X equals
π by two, the gradient of

the tangent is 0.

And it's also 0 - Π Pi 2, and at
three π by 2.

At other points, the symmetry of
the sine curve can help.

So the gradient of the tangent
at X equals 0 is positive.

And it must be equal to
the gradient of the

tangent at X equals 2π.

Also, the gradient of the
tangent at X equals minus. Pi

must be equal to the gradient of
the tangent at X equals π.

They must also both have the
same magnitude as the gradient

at X equals 0.

But be negative.

Let's now plot the gradients
of these tangents on a graph

directly underneath the sine
graph.

Now assume we can join up the
points with a smooth curve.

Once we don't know the values
of the peaks and the troughs,

what we've drawn looks
remarkably like the graph of

a cosine function.

So now we've got an idea of what
we're looking for. Let's

differentiate cynex from first
principles. Now there's three

things that we need to know.

The first is our definition of a
derivative Cy by DX equals the

limit as Delta X approaches zero
of F of X Plus Delta X

minus F of X.

Or divided by its Delta X?

The second is one of our
trigonometric identity's and

that sign C minus sign D.

Equals twice the cause of C
plus 3. / 2 multiplied by

the sign of C minus D
divided by two.

And the third is the limit.

As theater approaches zero, that
sign Theta divided by theater

equals 1. Now we can see
this from a table of values.

So if we have a look at theater.

When it's in radians.

And then calculate.

Sign theater And then
sign theater divided by theater.

We can have a look and see

what's happening. So our theater
in radians. Let's look at one

first of all.

The sign of
one is 0.84147.

So if we calculate sine
Theta divided by Sita, we

just get the same 0.84147.

Now let's make theater smaller.
Let's go to 0.1.

The sign of 0.1
is 0.0998 three and

so on.

Just it continued
on there, sorry.

OK, if we do sign
theater divided by Theta then

we get 0.99833 and again.

Continues on.

Let's make seat even
smaller now at 0.01.

The sign of Theta
is going to be

0.00999, and so on.

So sine Theta divided
by Sita equals 0.9998

three and so on.

And you can see that as Theta is
getting smaller and smaller and

tending to 0.

Then sign Theta divided by
theater is tending to one.

OK, let's carry on with

the calculation. So we have
our Y equals sign

X. And let's start by just
looking at the top part of our

formula. Our function of X Plus
Delta X minus a function of X.

Sign X is our function of X, so
our function of X Plus Delta X

is the sign of X Plus Delta X.

Minus R function of X, so minus

sign X. And this is where we
have our sign. See take away as

signed D. So we can rewrite
this as twice the cause.

Of C Plus D divided by
two. So it's X Plus Delta

X Plus X divided by two.

Multiplied by the sign of C
minus D divided by two.

So it's X Plus Delta X.

Minus X divided by two.

So this equals twice the cause.
X Plus X is 2X Plus Delta

X or divided by 2 multiplied by
the sign of X minus X is

0. So we just have Delta X
divided by two.

I'm just going to tidy this up a
little bit further, so we've got

two times the cause.

2 / 2 gives us just RX plus
Delta X over 2 multiplied by the

sign of Delta X over 2.

So now if we go back to
divide by DX.

Equals the limit as Delta
X approaches 0.

Of F of X Plus Delta X
minus F of X.

Which is this? So it's two calls
X plus Delta X over 2 multiplied

by the sign of Delta X over 2.

All divided by Delta X.

Now multiplying the numerator,
the top part of the fraction by

two is exactly the same as
dividing the denominator. The

bottom part by two.

So I'm going to rewrite this.

Limit of Delta X tends to 0.

Taking this too from here.

And.
Putting it as a division

in the denominator.

And I've done that so that you
can see here. We've got the sign

of Delta X over 2.

Divided by Delta X over 2
and that was where I third

fact came in that when we
had signed theater divided

by Sita when we took the
limit of Delta extending to

0, this will tend to one.

So let's actually take the
limit. Now is Delta X approaches

0 sign Delta X over 2 divided by
Delta X over 2?

Is one and Delta Rex over 2. As
Delta X approaches, zero will be

0. So we're just left
with our derivative of

sine X thing Cos X.

Let's have a look at the
derivative now from first

principles of Cos X.

The graph of Cos X looks
like this.

Again, we can draw the tangents
at certain points.

And we can plot their values on
a graph directly underneath the

cosine graph. This time if we
join the points with a smooth

curve, we get a graph that looks
like minus sign X.

Now before we
differentiate cause X

from first principles,
let's just have a look at

the graph of the sign of
X plus Π by 2.

The graph of sign of X plus Π by
2 looks like this.

It's just the graph of cynex
with everything happening

pie by two earlier.

It's as though we've shifted the
sign curve back along the X axis

by Π by 2.

And that just gives us Cossacks.

The two functions are the same.

Now shifting the sign curve back
along the Axis doesn't affect

the shape of the curve, so it
doesn't affect the shape of its

gradient function. Only the
position of the gradient

function on the X axis.

So the derivative of sign of X
plus Π by two is the cause of X

plus Π by 2.

But the graph of cause of X plus
Π by two is identical to the

graph of minus sign X.

So the derivative of Cos X.

Is minus sign X?

Let's have a look at the

calculation now. Y equals Cos

X. And we need to know three

things again. To help us do the
calculation, the first is our

definition of our derivative DY
by DX equals the limit as Delta

X approaches 0.

About function of X Plus
Delta X minus a function

of X divided by Delta X.

The second thing we need to know
is one of the trigonometric

identity's. And that cause see
minus calls D equals minus two

times the sign of C Plus D
divided by 2 multiplied by the

sign of C minus D divided by

two. And the third thing
is the limit. As theater

approaches 0. Sign Theta divided
by feature is equal to 1.

So let's have a look.

Add putting our function into
this top part against RF of X

Plus Delta X minus a function of

X. Is equal to.

I cause of X Plus

Delta X. Minus a function of
X which is called sex.

So here we have our cause of C
minus icons of D.

So let's do this
substitution now.

So it's minus twice the sign of
C Plus D over 2, so that's X

Plus Delta X.

Plus, X all divided by two.

Multiplied by the sign of C
minus T over 2. So that's X

Plus Delta X.

Minus X divided by two.

Which equals minus two times
the sign of X plus

X is 2X plus Delta
X divided by two.

Multiplied by the sign of X
minus X is 0, so we're just left

with Delta X divided by two.

And again, as we did before.

We write this is minus 2 sign 2X
divided by two leaves us with X

Plus our Delta X over 2
multiplied by our sign of Delta

X over 2.

Now we can turn the page and put

this into our.
Formula.

So that we have the why
by DX equals the limit a

Delta X approaches zero of minus
2 signs of X Plus Delta

X over 2.

Multiplied by sign Delta X over

2. All divided by Delta

X. I'm not equals, and again
we're going to do this change

where instead of multiplying the
numerator by two, I'm going to

divide the denominator by two.
So we have the limit as Delta X

approaches 0. Of minus
the sign of X Plus Delta

X divided by 2 multiplied
by the sign of Delta X

divided by two all
divided by Delta X

divided by two.

Now as we take the limit as
Delta X approaches 0. Again,

here we've got the sign of Delta
X over 2 divided by Delta X over

2, and we know that as Delta X
approaches zero, then this tends

to one. And then as Delta X
divided by two, here tends to 0.

Then our derivative will be
minus the sign of X.

So our derivative of Cos X is
minus sign X.