1
00:00:02,570 --> 00:00:04,916
The graph of Cynex looks like

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00:00:04,916 --> 00:00:09,580
this. And we can see the
gradient of the tangents at

3
00:00:09,580 --> 00:00:11,325
certain points of the graph.

4
00:00:12,340 --> 00:00:16,520
For instance, when X equals
π by two, the gradient of

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00:00:16,520 --> 00:00:18,040
the tangent is 0.

6
00:00:19,140 --> 00:00:24,194
And it's also 0 - Π Pi 2, and at
three π by 2.

7
00:00:25,310 --> 00:00:30,271
At other points, the symmetry of
the sine curve can help.

8
00:00:31,160 --> 00:00:35,888
So the gradient of the tangent
at X equals 0 is positive.

9
00:00:36,440 --> 00:00:40,060
And it must be equal to
the gradient of the

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00:00:40,060 --> 00:00:41,870
tangent at X equals 2π.

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00:00:42,970 --> 00:00:47,579
Also, the gradient of the
tangent at X equals minus. Pi

12
00:00:47,579 --> 00:00:53,026
must be equal to the gradient of
the tangent at X equals π.

13
00:00:54,410 --> 00:00:58,414
They must also both have the
same magnitude as the gradient

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00:00:58,414 --> 00:00:59,870
at X equals 0.

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00:01:00,530 --> 00:01:01,739
But be negative.

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00:01:03,850 --> 00:01:07,689
Let's now plot the gradients
of these tangents on a graph

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00:01:07,689 --> 00:01:09,434
directly underneath the sine
graph.

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00:01:16,230 --> 00:01:20,490
Now assume we can join up the
points with a smooth curve.

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00:01:21,500 --> 00:01:25,508
Once we don't know the values
of the peaks and the troughs,

20
00:01:25,508 --> 00:01:28,514
what we've drawn looks
remarkably like the graph of

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00:01:28,514 --> 00:01:29,516
a cosine function.

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00:01:31,440 --> 00:01:36,036
So now we've got an idea of what
we're looking for. Let's

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00:01:36,036 --> 00:01:39,100
differentiate cynex from first
principles. Now there's three

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00:01:39,100 --> 00:01:41,398
things that we need to know.

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00:01:42,040 --> 00:01:48,956
The first is our definition of a
derivative Cy by DX equals the

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00:01:48,956 --> 00:01:55,872
limit as Delta X approaches zero
of F of X Plus Delta X

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00:01:55,872 --> 00:01:58,000
minus F of X.

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00:01:58,620 --> 00:02:00,918
Or divided by its Delta X?

29
00:02:01,860 --> 00:02:06,936
The second is one of our
trigonometric identity's and

30
00:02:06,936 --> 00:02:10,320
that sign C minus sign D.

31
00:02:11,130 --> 00:02:18,342
Equals twice the cause of C
plus 3. / 2 multiplied by

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00:02:18,342 --> 00:02:23,751
the sign of C minus D
divided by two.

33
00:02:24,770 --> 00:02:28,310
And the third is the limit.

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00:02:28,880 --> 00:02:36,250
As theater approaches zero, that
sign Theta divided by theater

35
00:02:36,250 --> 00:02:42,130
equals 1. Now we can see
this from a table of values.

36
00:02:42,750 --> 00:02:46,150
So if we have a look at theater.

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00:02:46,660 --> 00:02:48,728
When it's in radians.

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00:02:50,060 --> 00:02:51,539
And then calculate.

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00:02:52,150 --> 00:02:58,639
Sign theater And then
sign theater divided by theater.

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00:02:59,570 --> 00:03:02,377
We can have a look and see

41
00:03:02,377 --> 00:03:07,022
what's happening. So our theater
in radians. Let's look at one

42
00:03:07,022 --> 00:03:08,066
first of all.

43
00:03:09,010 --> 00:03:15,178
The sign of
one is 0.84147.

44
00:03:16,040 --> 00:03:21,550
So if we calculate sine
Theta divided by Sita, we

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00:03:21,550 --> 00:03:24,305
just get the same 0.84147.

46
00:03:25,960 --> 00:03:29,389
Now let's make theater smaller.
Let's go to 0.1.

47
00:03:29,960 --> 00:03:37,288
The sign of 0.1
is 0.0998 three and

48
00:03:37,288 --> 00:03:39,120
so on.

49
00:03:40,190 --> 00:03:42,710
Just it continued
on there, sorry.

50
00:03:43,720 --> 00:03:50,540
OK, if we do sign
theater divided by Theta then

51
00:03:50,540 --> 00:03:53,950
we get 0.99833 and again.

52
00:03:54,510 --> 00:03:55,500
Continues on.

53
00:03:57,240 --> 00:04:01,648
Let's make seat even
smaller now at 0.01.

54
00:04:02,860 --> 00:04:09,756
The sign of Theta
is going to be

55
00:04:09,756 --> 00:04:13,204
0.00999, and so on.

56
00:04:14,820 --> 00:04:22,404
So sine Theta divided
by Sita equals 0.9998

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00:04:22,404 --> 00:04:26,196
three and so on.

58
00:04:27,150 --> 00:04:32,064
And you can see that as Theta is
getting smaller and smaller and

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00:04:32,064 --> 00:04:33,198
tending to 0.

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00:04:33,720 --> 00:04:37,820
Then sign Theta divided by
theater is tending to one.

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00:04:39,010 --> 00:04:42,830
OK, let's carry on with

62
00:04:42,830 --> 00:04:50,072
the calculation. So we have
our Y equals sign

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00:04:50,072 --> 00:04:56,520
X. And let's start by just
looking at the top part of our

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00:04:56,520 --> 00:05:02,045
formula. Our function of X Plus
Delta X minus a function of X.

65
00:05:03,580 --> 00:05:09,895
Sign X is our function of X, so
our function of X Plus Delta X

66
00:05:09,895 --> 00:05:13,263
is the sign of X Plus Delta X.

67
00:05:13,930 --> 00:05:16,751
Minus R function of X, so minus

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00:05:16,751 --> 00:05:22,006
sign X. And this is where we
have our sign. See take away as

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00:05:22,006 --> 00:05:28,524
signed D. So we can rewrite
this as twice the cause.

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00:05:29,030 --> 00:05:36,530
Of C Plus D divided by
two. So it's X Plus Delta

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00:05:36,530 --> 00:05:40,280
X Plus X divided by two.

72
00:05:40,910 --> 00:05:47,345
Multiplied by the sign of C
minus D divided by two.

73
00:05:48,060 --> 00:05:51,180
So it's X Plus Delta X.

74
00:05:51,860 --> 00:05:54,870
Minus X divided by two.

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00:05:55,950 --> 00:06:03,191
So this equals twice the cause.
X Plus X is 2X Plus Delta

76
00:06:03,191 --> 00:06:10,989
X or divided by 2 multiplied by
the sign of X minus X is

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00:06:10,989 --> 00:06:16,132
0. So we just have Delta X
divided by two.

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00:06:16,910 --> 00:06:21,264
I'm just going to tidy this up a
little bit further, so we've got

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00:06:21,264 --> 00:06:22,508
two times the cause.

80
00:06:23,500 --> 00:06:30,670
2 / 2 gives us just RX plus
Delta X over 2 multiplied by the

81
00:06:30,670 --> 00:06:33,538
sign of Delta X over 2.

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00:06:34,280 --> 00:06:39,690
So now if we go back to
divide by DX.

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00:06:40,190 --> 00:06:45,638
Equals the limit as Delta
X approaches 0.

84
00:06:46,630 --> 00:06:50,150
Of F of X Plus Delta X
minus F of X.

85
00:06:51,200 --> 00:06:57,990
Which is this? So it's two calls
X plus Delta X over 2 multiplied

86
00:06:57,990 --> 00:07:01,870
by the sign of Delta X over 2.

87
00:07:02,540 --> 00:07:05,410
All divided by Delta X.

88
00:07:06,930 --> 00:07:11,880
Now multiplying the numerator,
the top part of the fraction by

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00:07:11,880 --> 00:07:16,380
two is exactly the same as
dividing the denominator. The

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00:07:16,380 --> 00:07:18,180
bottom part by two.

91
00:07:18,690 --> 00:07:20,748
So I'm going to rewrite this.

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00:07:21,410 --> 00:07:24,336
Limit of Delta X tends to 0.

93
00:07:25,130 --> 00:07:28,380
Taking this too from here.

94
00:07:28,430 --> 00:07:36,015
And.
Putting it as a division

95
00:07:36,015 --> 00:07:38,316
in the denominator.

96
00:07:39,530 --> 00:07:43,604
And I've done that so that you
can see here. We've got the sign

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00:07:43,604 --> 00:07:45,059
of Delta X over 2.

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00:07:45,570 --> 00:07:50,358
Divided by Delta X over 2
and that was where I third

99
00:07:50,358 --> 00:07:54,348
fact came in that when we
had signed theater divided

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00:07:54,348 --> 00:07:58,737
by Sita when we took the
limit of Delta extending to

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00:07:58,737 --> 00:08:01,131
0, this will tend to one.

102
00:08:02,660 --> 00:08:06,906
So let's actually take the
limit. Now is Delta X approaches

103
00:08:06,906 --> 00:08:11,538
0 sign Delta X over 2 divided by
Delta X over 2?

104
00:08:12,150 --> 00:08:18,198
Is one and Delta Rex over 2. As
Delta X approaches, zero will be

105
00:08:18,198 --> 00:08:23,424
0. So we're just left
with our derivative of

106
00:08:23,424 --> 00:08:25,964
sine X thing Cos X.

107
00:08:27,270 --> 00:08:32,060
Let's have a look at the
derivative now from first

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00:08:32,060 --> 00:08:33,976
principles of Cos X.

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00:08:35,150 --> 00:08:37,694
The graph of Cos X looks
like this.

110
00:08:39,750 --> 00:08:43,224
Again, we can draw the tangents
at certain points.

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00:08:45,520 --> 00:08:49,972
And we can plot their values on
a graph directly underneath the

112
00:08:49,972 --> 00:08:55,120
cosine graph. This time if we
join the points with a smooth

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00:08:55,120 --> 00:08:58,970
curve, we get a graph that looks
like minus sign X.

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00:08:59,700 --> 00:09:02,136
Now before we
differentiate cause X

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00:09:02,136 --> 00:09:05,790
from first principles,
let's just have a look at

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00:09:05,790 --> 00:09:10,256
the graph of the sign of
X plus Π by 2.

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00:09:11,960 --> 00:09:17,134
The graph of sign of X plus Π by
2 looks like this.

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00:09:17,780 --> 00:09:21,758
It's just the graph of cynex
with everything happening

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00:09:21,758 --> 00:09:23,526
pie by two earlier.

120
00:09:24,650 --> 00:09:29,460
It's as though we've shifted the
sign curve back along the X axis

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00:09:29,460 --> 00:09:30,940
by Π by 2.

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00:09:33,130 --> 00:09:35,476
And that just gives us Cossacks.

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00:09:37,120 --> 00:09:39,520
The two functions are the same.

124
00:09:41,590 --> 00:09:46,122
Now shifting the sign curve back
along the Axis doesn't affect

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00:09:46,122 --> 00:09:51,478
the shape of the curve, so it
doesn't affect the shape of its

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00:09:51,478 --> 00:09:54,774
gradient function. Only the
position of the gradient

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00:09:54,774 --> 00:09:56,834
function on the X axis.

128
00:10:01,880 --> 00:10:06,936
So the derivative of sign of X
plus Π by two is the cause of X

129
00:10:06,936 --> 00:10:08,200
plus Π by 2.

130
00:10:09,630 --> 00:10:14,940
But the graph of cause of X plus
Π by two is identical to the

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00:10:14,940 --> 00:10:16,710
graph of minus sign X.

132
00:10:17,750 --> 00:10:20,480
So the derivative of Cos X.

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00:10:21,050 --> 00:10:22,710
Is minus sign X?

134
00:10:24,510 --> 00:10:27,228
Let's have a look at the

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00:10:27,228 --> 00:10:31,050
calculation now. Y equals Cos

136
00:10:31,050 --> 00:10:34,742
X. And we need to know three

137
00:10:34,742 --> 00:10:41,420
things again. To help us do the
calculation, the first is our

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00:10:41,420 --> 00:10:47,864
definition of our derivative DY
by DX equals the limit as Delta

139
00:10:47,864 --> 00:10:49,475
X approaches 0.

140
00:10:50,480 --> 00:10:55,400
About function of X Plus
Delta X minus a function

141
00:10:55,400 --> 00:10:58,352
of X divided by Delta X.

142
00:10:59,570 --> 00:11:03,890
The second thing we need to know
is one of the trigonometric

143
00:11:03,890 --> 00:11:10,640
identity's. And that cause see
minus calls D equals minus two

144
00:11:10,640 --> 00:11:17,712
times the sign of C Plus D
divided by 2 multiplied by the

145
00:11:17,712 --> 00:11:21,520
sign of C minus D divided by

146
00:11:21,520 --> 00:11:28,638
two. And the third thing
is the limit. As theater

147
00:11:28,638 --> 00:11:35,819
approaches 0. Sign Theta divided
by feature is equal to 1.

148
00:11:37,370 --> 00:11:39,380
So let's have a look.

149
00:11:40,160 --> 00:11:45,836
Add putting our function into
this top part against RF of X

150
00:11:45,836 --> 00:11:49,147
Plus Delta X minus a function of

151
00:11:49,147 --> 00:11:52,179
X. Is equal to.

152
00:11:53,420 --> 00:11:57,140
I cause of X Plus

153
00:11:57,140 --> 00:12:02,919
Delta X. Minus a function of
X which is called sex.

154
00:12:03,760 --> 00:12:08,212
So here we have our cause of C
minus icons of D.

155
00:12:09,310 --> 00:12:12,106
So let's do this
substitution now.

156
00:12:13,140 --> 00:12:20,475
So it's minus twice the sign of
C Plus D over 2, so that's X

157
00:12:20,475 --> 00:12:21,942
Plus Delta X.

158
00:12:22,620 --> 00:12:25,698
Plus, X all divided by two.

159
00:12:28,880 --> 00:12:35,549
Multiplied by the sign of C
minus T over 2. So that's X

160
00:12:35,549 --> 00:12:37,088
Plus Delta X.

161
00:12:37,590 --> 00:12:40,700
Minus X divided by two.

162
00:12:41,700 --> 00:12:48,530
Which equals minus two times
the sign of X plus

163
00:12:48,530 --> 00:12:54,677
X is 2X plus Delta
X divided by two.

164
00:12:54,680 --> 00:13:01,442
Multiplied by the sign of X
minus X is 0, so we're just left

165
00:13:01,442 --> 00:13:04,340
with Delta X divided by two.

166
00:13:04,340 --> 00:13:07,328
And again, as we did before.

167
00:13:07,330 --> 00:13:14,050
We write this is minus 2 sign 2X
divided by two leaves us with X

168
00:13:14,050 --> 00:13:19,426
Plus our Delta X over 2
multiplied by our sign of Delta

169
00:13:19,426 --> 00:13:20,770
X over 2.

170
00:13:21,890 --> 00:13:24,490
Now we can turn the page and put

171
00:13:24,490 --> 00:13:31,410
this into our.
Formula.

172
00:13:32,440 --> 00:13:39,808
So that we have the why
by DX equals the limit a

173
00:13:39,808 --> 00:13:47,176
Delta X approaches zero of minus
2 signs of X Plus Delta

174
00:13:47,176 --> 00:13:49,018
X over 2.

175
00:13:49,760 --> 00:13:52,952
Multiplied by sign Delta X over

176
00:13:52,952 --> 00:13:56,516
2. All divided by Delta

177
00:13:56,516 --> 00:14:02,752
X. I'm not equals, and again
we're going to do this change

178
00:14:02,752 --> 00:14:07,504
where instead of multiplying the
numerator by two, I'm going to

179
00:14:07,504 --> 00:14:13,120
divide the denominator by two.
So we have the limit as Delta X

180
00:14:13,120 --> 00:14:19,134
approaches 0. Of minus
the sign of X Plus Delta

181
00:14:19,134 --> 00:14:24,942
X divided by 2 multiplied
by the sign of Delta X

182
00:14:24,942 --> 00:14:29,166
divided by two all
divided by Delta X

183
00:14:29,166 --> 00:14:30,750
divided by two.

184
00:14:32,010 --> 00:14:36,606
Now as we take the limit as
Delta X approaches 0. Again,

185
00:14:36,606 --> 00:14:42,351
here we've got the sign of Delta
X over 2 divided by Delta X over

186
00:14:42,351 --> 00:14:47,330
2, and we know that as Delta X
approaches zero, then this tends

187
00:14:47,330 --> 00:14:53,374
to one. And then as Delta X
divided by two, here tends to 0.

188
00:14:54,090 --> 00:14:59,490
Then our derivative will be
minus the sign of X.

189
00:15:00,940 --> 00:15:06,310
So our derivative of Cos X is
minus sign X.