1
00:00:00,000 --> 00:00:04,490


2
00:00:04,490 --> 00:00:08,710
Let's prove with the derivative of e to the x's, and I think

3
00:00:08,710 --> 00:00:11,400
that this is one of the most amazing things, depending on

4
00:00:11,400 --> 00:00:14,060
how you view it about either calculus or math

5
00:00:14,060 --> 00:00:14,705
or the universe.

6
00:00:14,705 --> 00:00:17,219


7
00:00:17,219 --> 00:00:19,469
Well we're essentially going to prove-- I've already told you

8
00:00:19,469 --> 00:00:23,234
before that the derivative of e to the x is equal to e to

9
00:00:23,234 --> 00:00:24,599
the x, which is amazing.

10
00:00:24,600 --> 00:00:29,660
The slope at any point of that line is equal to the x value--

11
00:00:29,660 --> 00:00:33,049
is equal to the function at that point, not the x value.

12
00:00:33,049 --> 00:00:35,009
The slope at any point is equal to e.

13
00:00:35,009 --> 00:00:36,979
That is mind boggling.

14
00:00:36,979 --> 00:00:39,219
And that also means that the second derivative at any point

15
00:00:39,219 --> 00:00:42,109
is equal to the function of that value or the third

16
00:00:42,109 --> 00:00:46,519
derivative, or the infinite derivative, and that never

17
00:00:46,520 --> 00:00:47,380
ceases to amaze me.

18
00:00:47,380 --> 00:00:49,300
But anyway back to work.

19
00:00:49,299 --> 00:00:50,219
So how are we going to prove this?

20
00:00:50,219 --> 00:00:53,409
Well we already proved-- I actually just did it right

21
00:00:53,409 --> 00:00:57,629
before starting this video-- that the derivative-- and some

22
00:00:57,630 --> 00:00:59,770
people actually call this the definition of e.

23
00:00:59,770 --> 00:01:00,610
They go the other way around.

24
00:01:00,609 --> 00:01:02,519
They say there is some number for which this is true,

25
00:01:02,520 --> 00:01:04,140
and we call that number e.

26
00:01:04,140 --> 00:01:10,310
So it could almost be viewed as a little bit circular, but be

27
00:01:10,310 --> 00:01:14,769
we said that e is equal to the limit as n approaches infinity

28
00:01:14,769 --> 00:01:18,140
of 1 over 1 plus n to the end.

29
00:01:18,140 --> 00:01:21,930
And then using this we actually proved that derivative of

30
00:01:21,930 --> 00:01:25,710
ln of x is equal to 1/x.

31
00:01:25,709 --> 00:01:28,629
The derivative of log base e of x is equal to 1/x.

32
00:01:28,629 --> 00:01:33,060
So now that we prove this out, let's use this to prove this.

33
00:01:33,060 --> 00:01:38,310


34
00:01:38,310 --> 00:01:40,290
Let me keep switching colors to keep it interesting.

35
00:01:40,290 --> 00:01:46,820
Let's take the derivative of ln of e to the x.

36
00:01:46,819 --> 00:01:50,279


37
00:01:50,280 --> 00:01:51,420
This is almost trivial.

38
00:01:51,420 --> 00:01:57,010
This is equal to the logarithm of a to the b is equal to b

39
00:01:57,010 --> 00:01:59,200
times the logarithm of a, so this is equal to the

40
00:01:59,200 --> 00:02:04,600
derivative of x ln of e.

41
00:02:04,599 --> 00:02:07,640
And this is just saying e to what power is equal to e.

42
00:02:07,640 --> 00:02:09,650
Well, to the first power, right?

43
00:02:09,650 --> 00:02:13,730
So this just equals the derivative of x, which we

44
00:02:13,729 --> 00:02:14,840
have shown as equal to 1.

45
00:02:14,840 --> 00:02:17,479
I think we have shown it, hopefully we've shown it.

46
00:02:17,479 --> 00:02:20,649
If we haven't, that's actually a very easy one to prove.

47
00:02:20,650 --> 00:02:21,700
OK fair enough.

48
00:02:21,699 --> 00:02:23,629
We did that.

49
00:02:23,629 --> 00:02:25,030
But let's do this another way.

50
00:02:25,030 --> 00:02:28,280
Let's use the chain rule.

51
00:02:28,280 --> 00:02:29,419
So what doe the chain rule say?

52
00:02:29,419 --> 00:02:32,609
If we have f of g of x, where we have one function embedded

53
00:02:32,610 --> 00:02:35,660
in another one, the chain rule say we take the derivative of

54
00:02:35,659 --> 00:02:50,550
the inside function, so d/dx of e to the x.

55
00:02:50,550 --> 00:02:53,320
And then we take the derivative of the outside function or the

56
00:02:53,319 --> 00:02:56,340
derivative of the outside function with respect

57
00:02:56,340 --> 00:02:57,400
to the inner function.

58
00:02:57,400 --> 00:03:00,569
You can almost view it that way.

59
00:03:00,569 --> 00:03:03,000
So the derivative ln of e to the x with

60
00:03:03,000 --> 00:03:04,530
respect to e to the x.

61
00:03:04,530 --> 00:03:05,530
I know that's a little confusing.

62
00:03:05,530 --> 00:03:09,680
You could have written a d e to the x down over here, but I

63
00:03:09,680 --> 00:03:11,760
think you know the chain rule by now.

64
00:03:11,759 --> 00:03:15,649
That is equal to 1 over e to the x.

65
00:03:15,650 --> 00:03:18,055
And that just comes from this.

66
00:03:18,055 --> 00:03:19,939
But instead of an x, we have e to the x.

67
00:03:19,939 --> 00:03:23,340
So this is just a chain rule.

68
00:03:23,340 --> 00:03:24,830
Well what else do we know?

69
00:03:24,830 --> 00:03:28,090
We know that this is equal to this, and we also know that

70
00:03:28,090 --> 00:03:29,840
this is equal to this.

71
00:03:29,840 --> 00:03:31,909
So this must be equal to this.

72
00:03:31,909 --> 00:03:34,740
So this must be equal to 1.

73
00:03:34,740 --> 00:03:37,150
Well let's just multiply both sides of this

74
00:03:37,150 --> 00:03:41,159
equation by e to the x.

75
00:03:41,159 --> 00:03:43,069
We get on the left hand side, we're just left

76
00:03:43,069 --> 00:03:44,245
with this expression.

77
00:03:44,246 --> 00:03:51,450
The derivative of e to the x times- we're multiplying both

78
00:03:51,449 --> 00:03:54,089
sides by e to the x, times e to the x over e to the x.

79
00:03:54,090 --> 00:03:57,650
I just chose to put the e to the x on this term,

80
00:03:57,650 --> 00:03:59,740
is equal to e to the x.

81
00:03:59,740 --> 00:04:01,320
This is 1.

82
00:04:01,319 --> 00:04:02,669
Scratch it out.

83
00:04:02,669 --> 00:04:03,959
We're done.

84
00:04:03,960 --> 00:04:05,890
That might not have been completely satisfying

85
00:04:05,889 --> 00:04:07,639
for you, but it works.

86
00:04:07,639 --> 00:04:12,909
The derivative of e to the x is equal to e to the x.

87
00:04:12,909 --> 00:04:16,490


88
00:04:16,490 --> 00:04:21,259
I think the school or the nation should take a national

89
00:04:21,259 --> 00:04:23,250
holiday or something, and people should just

90
00:04:23,250 --> 00:04:27,620
ponder this, because it really is fascinating.

91
00:04:27,620 --> 00:04:31,730
But then actually this will lead us to I would say even

92
00:04:31,730 --> 00:04:36,200
more dramatic results in the not too far off future.

93
00:04:36,199 --> 00:04:37,870
Anyway, I'll see in the next video.

94
00:04:37,870 --> 00:04:39,800