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.