1
00:00:00,000 --> 00:00:00,790


2
00:00:00,790 --> 00:00:02,730
In the last presentation, I hopefully gave you a little

3
00:00:02,730 --> 00:00:05,290
bit of an intuition of what a derivative is.

4
00:00:05,290 --> 00:00:07,919
It's really just a way to find the slope at a given

5
00:00:07,919 --> 00:00:10,099
point along the curve.

6
00:00:10,099 --> 00:00:12,209
Now we'll actually apply it to some functions.

7
00:00:12,210 --> 00:00:18,200
So let's say I had the function f of x.

8
00:00:18,199 --> 00:00:22,149
f of x is equal to x squared.

9
00:00:22,149 --> 00:00:27,324
And I want to know what is the slope of this curve.

10
00:00:27,324 --> 00:00:40,409


11
00:00:40,409 --> 00:00:47,219
What is the slope at x is equal to-- let's say at x equals 3.

12
00:00:47,219 --> 00:00:48,519
What is the slope of x?

13
00:00:48,520 --> 00:00:51,720
Let's draw out what I'm asking.

14
00:00:51,719 --> 00:00:53,299
Coordinate axis.

15
00:00:53,299 --> 00:00:57,309
x-coordinate, that's the y-coordinate.

16
00:00:57,310 --> 00:00:59,260
And then if I were to draw-- let me pick a different color.

17
00:00:59,259 --> 00:01:05,909


18
00:01:05,909 --> 00:01:10,689
So we want to say what is the slope when x is equal to 3.

19
00:01:10,689 --> 00:01:16,310


20
00:01:16,310 --> 00:01:19,590
This is x equals 3.

21
00:01:19,590 --> 00:01:23,240
And of course when x equals 3, f of x is equal to 9.

22
00:01:23,239 --> 00:01:25,009
We know that, right?

23
00:01:25,010 --> 00:01:28,829


24
00:01:28,829 --> 00:01:32,000
So what we do is we take a point, maybe a little bit

25
00:01:32,000 --> 00:01:33,120
further along the curve.

26
00:01:33,120 --> 00:01:37,880
Let's say this point right here is 3 plus h.

27
00:01:37,879 --> 00:01:39,780
And I keep it abstract as h because as you know we're

28
00:01:39,780 --> 00:01:41,719
going to take the limit as h approaches 0.

29
00:01:41,719 --> 00:01:45,230
And at this point right here is what?

30
00:01:45,230 --> 00:01:48,390
It's 3 plus h squared, right?

31
00:01:48,390 --> 00:01:52,000
Because the function is f of x is equal to x squared.

32
00:01:52,000 --> 00:02:08,699
So this point right here is 3 plus h, 3 plus h squared.

33
00:02:08,699 --> 00:02:10,908
Because we just take the 3 plus h and put it into x squared

34
00:02:10,908 --> 00:02:12,359
and we get 3 plus h squared.

35
00:02:12,360 --> 00:02:17,000
And this point here is of course 3, 9.

36
00:02:17,000 --> 00:02:19,340
What we want to do is we want to find the slope

37
00:02:19,340 --> 00:02:20,539
between these two point.

38
00:02:20,539 --> 00:02:23,599


39
00:02:23,599 --> 00:02:25,310
I really have to find a better tool.

40
00:02:25,310 --> 00:02:28,890
This one keeps freezing, I think it's too CPU intensive.

41
00:02:28,889 --> 00:02:29,500
But anyway.

42
00:02:29,500 --> 00:02:31,479
So we want to find the slope between these two points.

43
00:02:31,479 --> 00:02:34,500
So what's the slope? so it's a change in y, so it's 3 plus h

44
00:02:34,500 --> 00:02:50,240
squared minus this y minus 9 over the change in x.

45
00:02:50,240 --> 00:03:03,000
Well that's 3 plus h minus 3.

46
00:03:03,000 --> 00:03:05,780
So if we simplify this top part or we multiply it out,

47
00:03:05,780 --> 00:03:06,719
what's 3 plus h squared?

48
00:03:06,719 --> 00:03:19,259
That's 9 plus 6h plus h squared, and then get the minus

49
00:03:19,259 --> 00:03:26,519
9, and all of that is over-- well this 3 and this minus

50
00:03:26,520 --> 00:03:29,730
3 cancel out, so all you're left is with h.

51
00:03:29,729 --> 00:03:31,889
And even if we simplify this, this 9 minus

52
00:03:31,889 --> 00:03:34,891
9, they cancel out.

53
00:03:34,891 --> 00:03:37,004
So let me go up here.

54
00:03:37,004 --> 00:03:40,699


55
00:03:40,699 --> 00:03:49,209
We're left with-- this pen keeps freezing-- it's 6h

56
00:03:49,210 --> 00:03:53,040
plus h squared over h.

57
00:03:53,039 --> 00:03:55,500
And now we would simplify this, right, because we can divide

58
00:03:55,500 --> 00:03:57,159
the top and the bottom, that numerator and the

59
00:03:57,159 --> 00:03:58,259
denominator by h.

60
00:03:58,259 --> 00:04:05,199
And you get 6 plus h squared.

61
00:04:05,199 --> 00:04:08,229
So that's the slope between these two points.

62
00:04:08,229 --> 00:04:10,239
It's 6 plus h squared.

63
00:04:10,240 --> 00:04:12,980
So if we want to find the instantaneous slope at the

64
00:04:12,979 --> 00:04:17,560
point x equals 3, f of x is equal to 9, or the point 3,9,

65
00:04:17,560 --> 00:04:20,449
we just have to find the limit as h approaches 0 here.

66
00:04:20,449 --> 00:04:27,670
So we'll just take the limit as h approaches 0.

67
00:04:27,670 --> 00:04:29,240
Well this is an easy limit problem, right?

68
00:04:29,240 --> 00:04:32,620
What's the limit of 6 plus h squared as h approaches 0?

69
00:04:32,620 --> 00:04:34,340
Well it equals 6.

70
00:04:34,339 --> 00:04:38,299
So we now know that the slope of this curve at the

71
00:04:38,300 --> 00:04:41,970
point x equals 3 is 6.

72
00:04:41,970 --> 00:04:45,860


73
00:04:45,860 --> 00:04:49,600
So if you actually did a traditional rise over run,

74
00:04:49,600 --> 00:04:52,390
the slope, this change in y over change in x is 6.

75
00:04:52,389 --> 00:04:54,829
So we have the instantaneous slope at exactly the

76
00:04:54,829 --> 00:04:57,139
point x is equal to 3.

77
00:04:57,139 --> 00:05:00,069
So that's useful.

78
00:05:00,069 --> 00:05:05,959
You know if this was a graph of someone's position, we would

79
00:05:05,959 --> 00:05:08,569
then know kind of the instantaneous velocity, which

80
00:05:08,569 --> 00:05:09,819
is-- well I won't go into that.

81
00:05:09,819 --> 00:05:11,709
I'll do a separate module on physics.

82
00:05:11,709 --> 00:05:13,620
But this was useful, but let's see if we can do more

83
00:05:13,620 --> 00:05:16,439
generalized version where we don't have to know ahead of

84
00:05:16,439 --> 00:05:18,550
time what point we want to find the slope at.

85
00:05:18,550 --> 00:05:22,290
If we can get a generalized formula for the slope at any

86
00:05:22,290 --> 00:05:26,870
point along the graph f of x is equal to x squared.

87
00:05:26,870 --> 00:05:30,139
So let me clear this.

88
00:05:30,139 --> 00:05:42,469
So we're going to stick with f of x is equal to x squared.

89
00:05:42,470 --> 00:05:46,180
And we know that the slope at any point of this is just going

90
00:05:46,180 --> 00:06:03,050
to be the limit as h approaches 0 of f of x plus h

91
00:06:03,050 --> 00:06:07,530
minus f of access.

92
00:06:07,529 --> 00:06:11,649
All of that over h.

93
00:06:11,649 --> 00:06:14,589
This part right here, this is just the slope formula that

94
00:06:14,589 --> 00:06:16,079
you learned years ago.

95
00:06:16,079 --> 00:06:18,609
It's just change in y over change in x.

96
00:06:18,610 --> 00:06:21,449
And all we're doing is we're seeing what happens as the

97
00:06:21,449 --> 00:06:23,860
change in x gets smaller and smaller and smaller as it

98
00:06:23,860 --> 00:06:24,860
actually approaches 0.

99
00:06:24,860 --> 00:06:27,090
And that's why we can get the instantaneous change at

100
00:06:27,089 --> 00:06:28,250
that point in the curve.

101
00:06:28,250 --> 00:06:30,610
So let's apply this definition of a derivative

102
00:06:30,610 --> 00:06:33,490
to this function.

103
00:06:33,490 --> 00:06:37,040
And actually if you want to know the notation, I think

104
00:06:37,040 --> 00:06:39,189
this is the notation Lagrange came up with.

105
00:06:39,189 --> 00:06:43,949
This is equal to f prime of x.

106
00:06:43,949 --> 00:06:45,209
Don't take my word on it on Lagrange.

107
00:06:45,209 --> 00:06:46,969
You might want to look it up on Wikipedia.

108
00:06:46,970 --> 00:06:48,250
But this [UNINTELLIGIBLE]

109
00:06:48,250 --> 00:06:51,980
derivative of f of x is f prime of x.

110
00:06:51,980 --> 00:06:53,629
Let's apply it to x squared.

111
00:06:53,629 --> 00:06:59,490
So we're going to say the limit as h approaches

112
00:06:59,490 --> 00:07:02,790
0 of f of x plus h.

113
00:07:02,790 --> 00:07:08,210
Well, f of x plus h is just-- this pen driving me

114
00:07:08,209 --> 00:07:13,579
crazy-- x plus h squared.

115
00:07:13,579 --> 00:07:16,689
I just took the x plus h and put it into f of x.

116
00:07:16,689 --> 00:07:24,730
Minus f of x--well that's just x squared-- over h.

117
00:07:24,730 --> 00:07:30,490
And this is equal to the limit as h approaches 0.

118
00:07:30,490 --> 00:07:32,240
Just multiply this out of.

119
00:07:32,240 --> 00:07:43,710
x squared plus 2xh plus h squared minus x squared--

120
00:07:43,709 --> 00:07:48,349
running out of space-- all of that over h.

121
00:07:48,350 --> 00:07:49,310
Let's simplify this.

122
00:07:49,310 --> 00:07:53,160
This x squared cancels out with this minus x squared.

123
00:07:53,160 --> 00:07:56,260
And then we can divide the numerator and the denominator

124
00:07:56,259 --> 00:08:07,004
by h, and we're left with the limit as h approaches 0.

125
00:08:07,004 --> 00:08:14,180
Numerator and denominator by h of 2x plus h.

126
00:08:14,180 --> 00:08:16,470
Well this is easy.

127
00:08:16,470 --> 00:08:19,520
This goes to 0, this is just equal to 2x.

128
00:08:19,519 --> 00:08:20,639
So there we have it.

129
00:08:20,639 --> 00:08:24,519
The limit as h approaches 0 is equal to 2x.

130
00:08:24,519 --> 00:08:27,649
And this is equal to f prime of x, so the derivative of f of

131
00:08:27,649 --> 00:08:32,414
x, which is the denoted by f prime of x is equal to 2x.

132
00:08:32,414 --> 00:08:33,659
Well what does it tell us?

133
00:08:33,659 --> 00:08:35,269
What have we done for ourselves?

134
00:08:35,269 --> 00:08:38,929
Well now I can give you any point along the curve.

135
00:08:38,929 --> 00:08:43,463
Let's say we want to know the slope at the point 16, right.

136
00:08:43,464 --> 00:08:46,260


137
00:08:46,259 --> 00:08:51,200
When at the point 16,256.

138
00:08:51,200 --> 00:08:53,870
That's a point along f of x equals x squared.

139
00:08:53,870 --> 00:08:55,870
It's just 16 and then 16 squared.

140
00:08:55,870 --> 00:08:57,250
What's the slope at that point?

141
00:08:57,250 --> 00:08:59,745
Well we now know the slope is 2 times 16.

142
00:08:59,745 --> 00:09:04,639


143
00:09:04,639 --> 00:09:09,340
So the slope is equal to 32.

144
00:09:09,340 --> 00:09:12,545
Whatever the x value is you just put into this f prime of

145
00:09:12,544 --> 00:09:15,750
x function or the derivative function, and you'll get

146
00:09:15,750 --> 00:09:17,059
the slope at that point.

147
00:09:17,059 --> 00:09:19,269
I think that's pretty neat and I'll show you how in future

148
00:09:19,269 --> 00:09:22,100
presentations how we can apply this to physics and

149
00:09:22,100 --> 00:09:24,090
optimization problems and a whole other set of things.

150
00:09:24,090 --> 00:09:26,290
And I'm also going to show you how to find the derivatives for

151
00:09:26,289 --> 00:09:28,179
a whole set of other functions.

152
00:09:28,179 --> 00:09:29,029
I'll see in the next presentation.

153
00:09:29,029 --> 00:09:31,000