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PROFESSOR: All right, so
let's begin Lecture Six.
10
00:00:27,06 --> 00:00:44,63
We're talking today about
exponentials and logarithms.
11
00:00:44,63 --> 00:00:48,12
And these are the last
functions that I need to
12
00:00:48,12 --> 00:00:52,17
introduce, the last standard
functions that we need to
13
00:00:52,17 --> 00:00:55,11
connect with Calculus, that
you've learned about.
14
00:00:55,11 --> 00:00:58,23
And they're certainly as
fundamental, if not more so,
15
00:00:58,23 --> 00:01:00,83
than trigonometric functions.
16
00:01:00,83 --> 00:01:05,38
So first of all, we'll start
out with a number, a, which
17
00:01:05,38 --> 00:01:09,03
is positive, which is
usually called a base.
18
00:01:09,03 --> 00:01:13,3
And then we have these
properties that a to the
19
00:01:13,3 --> 00:01:14,87
power 0 is always 1.
20
00:01:14,87 --> 00:01:17,18
That's how we get started.
21
00:01:17,18 --> 00:01:21,27
And a^1 is a.
22
00:01:21,27 --> 00:01:24,03
And of course a^2 ,
not surprisingly,
23
00:01:24,03 --> 00:01:26,2
is a times a, etc.
24
00:01:26,2 --> 00:01:36,97
And the general rule is that
a^(X1 X2) is a^X1 times a^X2 .
25
00:01:36,97 --> 00:01:41,44
So this is the basic rule of
exponents, and with these two
26
00:01:41,44 --> 00:01:45,37
initial properties,
that defines the
27
00:01:45,37 --> 00:01:49,07
exponential function.
28
00:01:49,07 --> 00:01:53,77
And then there's an additional
property, which is deduced from
29
00:01:53,77 --> 00:01:59,02
these, which is the composition
of exponential functions, which
30
00:01:59,02 --> 00:02:03,33
is that you take a to the
X1 power, to the X2 power.
31
00:02:03,33 --> 00:02:08,39
Then that turns out to be
a to the X1 times X2.
32
00:02:08,39 --> 00:02:11,07
So that's an additional
property that we'll take
33
00:02:11,07 --> 00:02:14,14
for granted, which you
learned in high school.
34
00:02:14,14 --> 00:02:22,65
Now, in order to understand
what all the values of a^x are,
35
00:02:22,65 --> 00:02:28,62
we need to first remember that
if you're taking a rational
36
00:02:28,62 --> 00:02:34,35
power that it's the ratio of
two integers power of a.
37
00:02:34,35 --> 00:02:36,95
That's going to be a ^ m, and
then we're want to have to
38
00:02:36,95 --> 00:02:39,44
take the nth root of that.
39
00:02:39,44 --> 00:02:40,84
So that's the definition.
40
00:02:40,84 --> 00:02:50,34
And then, when you're defining
a ^ x, so a^x is defined
41
00:02:50,34 --> 00:03:00,37
for all x by filling in.
42
00:03:00,37 --> 00:03:03,21
So I'm gonna use that
expression in quotation
43
00:03:03,21 --> 00:03:09,93
marks, "filling
in" by continuity.
44
00:03:09,93 --> 00:03:12,63
This is really what your
calculator does when it gives
45
00:03:12,63 --> 00:03:16,45
you a to the power x, because
you can't even punch in
46
00:03:16,45 --> 00:03:17,54
the square root of x.
47
00:03:17,54 --> 00:03:19,59
It doesn't really exist
on your calculator.
48
00:03:19,59 --> 00:03:21,23
There's some decimal expansion.
49
00:03:21,23 --> 00:03:24,4
So it takes the decimal
expansion to a certain length
50
00:03:24,4 --> 00:03:26,6
and spits out a number
which is pretty close
51
00:03:26,6 --> 00:03:28,12
to the correct answer.
52
00:03:28,12 --> 00:03:32,17
But indeed, in theory, there is
an a to the power, square root
53
00:03:32,17 --> 00:03:34,74
of 2, even though the square
root of 2 is irrational.
54
00:03:34,74 --> 00:03:37,54
And there's a to the
pi and so forth.
55
00:03:37,54 --> 00:03:41,18
All right, so that's the
exponential function, and
56
00:03:41,18 --> 00:03:46,83
let's draw a picture of one.
57
00:03:46,83 --> 00:03:52,23
So we'll try, say y = 2^X here.
58
00:03:52,23 --> 00:03:56,26
And I'm not going to draw such
a careful graph, but let's just
59
00:03:56,26 --> 00:04:01,36
plot the most important point,
which is the point (0,1).
60
00:04:01,36 --> 00:04:04,51
That's 2^0, which is 1.
61
00:04:04,51 --> 00:04:08,94
And then maybe we'll go
back up here to - 1 here.
62
00:04:08,94 --> 00:04:13,86
And 2 to the - 1 is
this point here.
63
00:04:13,86 --> 00:04:16,81
This is (-1, 0.5) .
64
00:04:16,81 --> 00:04:18,99
The reciprocal.
65
00:04:18,99 --> 00:04:22,34
And over here, we have
1, and so that goes
66
00:04:22,34 --> 00:04:23,46
all the way up to 2.
67
00:04:23,46 --> 00:04:26,87
And then exponentials
are remarkably fast.
68
00:04:26,87 --> 00:04:30,68
So it's off the board what
happens next out at 2.
69
00:04:30,68 --> 00:04:36,48
It's already above my range
here, but the graph looks
70
00:04:36,48 --> 00:04:37,97
something like this.
71
00:04:37,97 --> 00:04:38,61
All right.
72
00:04:38,61 --> 00:04:42,12
Now I've just visually, at
least, graphically filled in
73
00:04:42,12 --> 00:04:43,24
all the rest of the points.
74
00:04:43,24 --> 00:04:46,73
You have to imagine all these
rational numbers, and so forth.
75
00:04:46,73 --> 00:04:51,67
So this point here would
have been (1, 2).
76
00:04:51,67 --> 00:04:53,33
And so forth.
77
00:04:53,33 --> 00:04:54,61
All right?
78
00:04:54,61 --> 00:05:01,04
So that's not too far along.
79
00:05:01,04 --> 00:05:01,95
So now what's our goal?
80
00:05:01,95 --> 00:05:04,02
Well, obviously we want
to do calculus here.
81
00:05:04,02 --> 00:05:08,05
So our goal, here, for now -
and it's gonna take a while.
82
00:05:08,05 --> 00:05:10,22
We have to think about
it pretty hard.
83
00:05:10,22 --> 00:05:22,25
We have to calculate what
this derivative is.
84
00:05:22,25 --> 00:05:26,02
All right, so we'll
get started.
85
00:05:26,02 --> 00:05:29,47
And the way we get started is
simply by plugging in the
86
00:05:29,47 --> 00:05:31,19
definition of the derivative.
87
00:05:31,19 --> 00:05:37,99
The derivative is the limit as
delta x goes to 0 of a to the
88
00:05:37,99 --> 00:05:45,08
x plus delta x, minus a to
the x, divided by delta x.
89
00:05:45,08 --> 00:05:50,32
So that's what it is.
90
00:05:50,32 --> 00:05:56,48
And now, the only step that we
can really perform here to make
91
00:05:56,48 --> 00:06:01,66
this is into something a little
bit simpler is to use this very
92
00:06:01,66 --> 00:06:03,2
first rule that we have here.
93
00:06:03,2 --> 00:06:06,93
That the exponential of
the sum is the product
94
00:06:06,93 --> 00:06:08,07
of the exponentials.
95
00:06:08,07 --> 00:06:10,34
So we have here, a^x .
96
00:06:10,34 --> 00:06:15,47
So what I want to use is
just the property that a^x
97
00:06:15,47 --> 00:06:22,46
delta x = (a^x) (a^delta x).
98
00:06:22,46 --> 00:06:26,88
And if I do that, I see that I
can factor out a common factor
99
00:06:26,88 --> 00:06:29,76
in the numerator, which is a^x.
100
00:06:29,76 --> 00:06:36,07
So we'll write this as the
limit as delta x goes to 0, of
101
00:06:36,07 --> 00:06:40,76
a to the x times this ratio,
now a to the delta x, minus
102
00:06:40,76 --> 00:06:49,28
1, divided by delta x.
103
00:06:49,28 --> 00:06:50,16
So far, so good?
104
00:06:50,16 --> 00:06:53,64
We're actually almost to
some serious progress here.
105
00:06:53,64 --> 00:06:58,65
So there's one other important
conceptual step which
106
00:06:58,65 --> 00:07:00,1
we need to understand.
107
00:07:00,1 --> 00:07:03,01
And this is a
relatively simple one.
108
00:07:03,01 --> 00:07:05,34
We actually did this
before, by the way.
109
00:07:05,34 --> 00:07:08,2
We did this with
sines and cosines.
110
00:07:08,2 --> 00:07:11,27
The next thing I want to point
out to you is that you're used
111
00:07:11,27 --> 00:07:15,68
to thinking of x as
being the variable.
112
00:07:15,68 --> 00:07:18,59
And indeed, already we were
discussing x as being the
113
00:07:18,59 --> 00:07:20,13
variable and a as being fixed.
114
00:07:20,13 --> 00:07:23,29
But for the purposes of this
limit, there's a different
115
00:07:23,29 --> 00:07:27,49
variable that's moving. x is
fixed and delta x is the
116
00:07:27,49 --> 00:07:29,34
thing that's moving.
117
00:07:29,34 --> 00:07:32,47
So that means that this factor
here, which is a common
118
00:07:32,47 --> 00:07:34,89
factor, is constant.
119
00:07:34,89 --> 00:07:36,59
And we can just factor
it out of the limit.
120
00:07:36,59 --> 00:07:39,45
It doesn't affect
the limit at all.
121
00:07:39,45 --> 00:07:42,69
A constant times a limit is the
same as whether we multiply
122
00:07:42,69 --> 00:07:44,58
before or after we
take the limit.
123
00:07:44,58 --> 00:07:46,83
So I'm just going to
factor that out.
124
00:07:46,83 --> 00:07:49,11
So that's my next step here.
125
00:07:49,11 --> 00:07:51,54
a^x, and then I have the limit.
126
00:07:51,54 --> 00:07:56,21
Delta x goes to 0 of a
to the delta x minus
127
00:07:56,21 --> 00:07:59,83
1, divided by delta x.
128
00:07:59,83 --> 00:08:02,18
All right?
129
00:08:02,18 --> 00:08:04,62
And so what I have here,
so this is by definition
130
00:08:04,62 --> 00:08:05,22
the derivative.
131
00:08:05,22 --> 00:08:09,29
So here is d/ dx of
a^x, and it's equal to
132
00:08:09,29 --> 00:08:12,73
this expression here.
133
00:08:12,73 --> 00:08:19,79
Now, I want to stare at this
expression, and see what it's
134
00:08:19,79 --> 00:08:23,69
telling us, because it's
telling us as much as we can
135
00:08:23,69 --> 00:08:27,68
get so far, without some...
136
00:08:27,68 --> 00:08:34,81
So first let's just look
at what this says.
137
00:08:34,81 --> 00:08:40,26
So what it's saying is that the
derivative of a^x is a^x times
138
00:08:40,26 --> 00:08:42,82
something that we
don't yet know.
139
00:08:42,82 --> 00:08:44,69
And I'm going to call
this something, this
140
00:08:44,69 --> 00:08:47,13
mystery number, M(a) .
141
00:08:47,13 --> 00:08:53,165
So I'm gonna make the label,
M(a) is equal to the limit as
142
00:08:53,165 --> 00:08:57,24
delta x goes to 0 of a to
the delta x minus 1
143
00:08:57,24 --> 00:09:00,01
divided by delta x.
144
00:09:00,01 --> 00:09:00,3
All right?
145
00:09:00,3 --> 00:09:08,87
So this is a definition.
146
00:09:08,87 --> 00:09:13,65
So this mystery number
M(a) has a geometric
147
00:09:13,65 --> 00:09:16
interpretation, as well.
148
00:09:16 --> 00:09:17,61
So let me describe that.
149
00:09:17,61 --> 00:09:19,19
It has a geometric
interpretation, and it's a
150
00:09:19,19 --> 00:09:20,63
very, very significant number.
151
00:09:20,63 --> 00:09:22,2
So let's work out what that is.
152
00:09:22,2 --> 00:09:25,69
So first of all, let's rewrite
the expression in the box,
153
00:09:25,69 --> 00:09:28,47
using the shorthand
for this number.
154
00:09:28,47 --> 00:09:32,93
So if I just rewrite it, it
says d/dx of a^x is equal
155
00:09:32,93 --> 00:09:37,8
to this factor, which
is M(a), times a^x .
156
00:09:37,8 --> 00:09:42,94
So the derivative of the
exponential is this
157
00:09:42,94 --> 00:09:44,79
mystery number times a^x.
158
00:09:44,79 --> 00:09:48,93
So we've almost solved the
problem of finding the
159
00:09:48,93 --> 00:09:50,56
derivative of a^x.
160
00:09:50,56 --> 00:09:53,27
We just have to figure out
this one number, M(a),
161
00:09:53,27 --> 00:09:55,72
and we get the rest.
162
00:09:55,72 --> 00:10:01,78
So let me point out two more
things about this number, M(a).
163
00:10:01,78 --> 00:10:10,21
So first of all, if I plug in
x = 0, that's going to be
164
00:10:10,21 --> 00:10:14,25
d / dx of a^x , at x = 0.
165
00:10:14,25 --> 00:10:19,15
According to this formula,
that's M(a) times a^0 ,
166
00:10:19,15 --> 00:10:21,37
which of course M(a).
167
00:10:21,37 --> 00:10:23,54
So what is M(a) ?
168
00:10:23,54 --> 00:10:26,41
M(a) is the derivative
of this function at 0.
169
00:10:26,41 --> 00:10:39,79
So M(a) is the slope of a^x
at x = 0, of the graph.
170
00:10:39,79 --> 00:10:41,33
The graph of a^x at 0.
171
00:10:41,33 --> 00:10:46,17
So again over here, if you
looked at the picture.
172
00:10:46,17 --> 00:10:48,45
I'll draw the one tangent
line in here, which
173
00:10:48,45 --> 00:10:50,64
is this one here.
174
00:10:50,64 --> 00:11:00,05
And this thing has slope,
what we're calling M(2).
175
00:11:00,05 --> 00:11:02,94
So, if I graph the function
y = 2 ^x, I'll get a
176
00:11:02,94 --> 00:11:03,74
certain slope here.
177
00:11:03,74 --> 00:11:05,91
If I graph it with a
different base, I might
178
00:11:05,91 --> 00:11:07,59
get another slope.
179
00:11:07,59 --> 00:11:13,39
And what we got so far is the
following phenomenon: if we
180
00:11:13,39 --> 00:11:16,31
know this one number, if we
know the slope at this one
181
00:11:16,31 --> 00:11:18,53
place, we will be able to
figure out the formula for
182
00:11:18,53 --> 00:11:23,32
the slope everywhere else.
183
00:11:23,32 --> 00:11:26,17
Now, that's actually exactly
the same thing that we did
184
00:11:26,17 --> 00:11:28,04
for sines and cosines.
185
00:11:28,04 --> 00:11:32,13
We knew the slope of the
sine and the cosine
186
00:11:32,13 --> 00:11:35,9
function at x = 0.
187
00:11:35,9 --> 00:11:37,45
The sine function had slope 1.
188
00:11:37,45 --> 00:11:39,47
The cosine function
had slope 0.
189
00:11:39,47 --> 00:11:42,93
And then from the sum formulas,
well that's exactly this kind
190
00:11:42,93 --> 00:11:44,92
of thing here, from
the sum formulas.
191
00:11:44,92 --> 00:11:47,11
This sum formula, in fact
is easier than the ones
192
00:11:47,11 --> 00:11:49,32
for sines and cosines.
193
00:11:49,32 --> 00:11:51,71
From the sum formulas,
we worked out what the
194
00:11:51,71 --> 00:11:53,62
slope was everywhere.
195
00:11:53,62 --> 00:11:57,61
So we're following the same
procedure that we did before.
196
00:11:57,61 --> 00:12:00,96
But at this point we're stuck.
197
00:12:00,96 --> 00:12:05,59
We're stuck, because that time
using radians, this very clever
198
00:12:05,59 --> 00:12:08,33
idea of radians in geometry, we
were able to actually figure
199
00:12:08,33 --> 00:12:09,64
out what the slope is.
200
00:12:09,64 --> 00:12:12,55
Whereas here, we're not
so sure, what M(2)
201
00:12:12,55 --> 00:12:14,92
is, for instance.
202
00:12:14,92 --> 00:12:17,2
We just don't know yet.
203
00:12:17,2 --> 00:12:21,65
So, the basic question that
we have to deal with right
204
00:12:21,65 --> 00:12:32,06
now is what is M(a)?
205
00:12:32,06 --> 00:12:34,68
That's what we're left with.
206
00:12:34,68 --> 00:12:42,62
And, the curious fact is
that the clever thing to
207
00:12:42,62 --> 00:12:51,26
do is to beg the question.
208
00:12:51,26 --> 00:12:54,73
So we're going to go through
a very circular route here.
209
00:12:54,73 --> 00:12:56,58
That is circuitous,
not circular.
210
00:12:56,58 --> 00:12:58,36
Circular is a bad word in math.
211
00:12:58,36 --> 00:13:00,475
That means that one thing
depends on another, and
212
00:13:00,475 --> 00:13:03,22
that depends on it, and
maybe both are wrong.
213
00:13:03,22 --> 00:13:05,4
Circuitous means, we're
going to be taking
214
00:13:05,4 --> 00:13:07,46
a roundabout route.
215
00:13:07,46 --> 00:13:10,61
And we're going to discover
that even though we refuse to
216
00:13:10,61 --> 00:13:12,58
answer this question right
now, we'll succeed in
217
00:13:12,58 --> 00:13:14,89
answering it eventually.
218
00:13:14,89 --> 00:13:15,34
All right?
219
00:13:15,34 --> 00:13:18,34
So how are we going
to beg the question?
220
00:13:18,34 --> 00:13:22,35
What we're going to say instead
is we're going to define a
221
00:13:22,35 --> 00:13:38,72
mystery base, or number e, as
the unique number,
222
00:13:38,72 --> 00:13:45,79
so that M(e) = 1.
223
00:13:45,79 --> 00:13:47,93
That's the trick that
we're going to use.
224
00:13:47,93 --> 00:13:50,98
We don't yet know what e is,
but we're just going to
225
00:13:50,98 --> 00:13:53,9
suppose that we have it.
226
00:13:53,9 --> 00:13:57,34
Now, I'm going to show you a
bunch of consequences of this,
227
00:13:57,34 --> 00:14:00,54
and also I have to persuade you
that it actually does exist.
228
00:14:00,54 --> 00:14:03,64
So first, let me explain what
the first consequence is.
229
00:14:03,64 --> 00:14:07,12
First of all, if M(e) is 1,
then if you look at this
230
00:14:07,12 --> 00:14:10,41
formula over here and you write
it down for e, you have
231
00:14:10,41 --> 00:14:13,72
something which is a very
usable formula. d / dx
232
00:14:13,72 --> 00:14:19,93
of e^x is just e^x.
233
00:14:19,93 --> 00:14:22,75
All right, so that's an
incredibly important formula
234
00:14:22,75 --> 00:14:24,54
which is the fundamental one.
235
00:14:24,54 --> 00:14:26,71
It's the only one you have to
remember from what we've done.
236
00:14:26,71 --> 00:14:29
So maybe I should have
highlighted it in
237
00:14:29 --> 00:14:34,76
several colors here.
238
00:14:34,76 --> 00:14:37,8
That's a big deal.
239
00:14:37,8 --> 00:14:40,63
Very happy.
240
00:14:40,63 --> 00:14:43,86
And again, let me just
emphasize, also that this
241
00:14:43,86 --> 00:14:52,09
is the one which at
x = 0 has slope 1.
242
00:14:52,09 --> 00:14:53,66
That's the way we
defined it, alright?
243
00:14:53,66 --> 00:15:00,64
So if you plug in x = 0 here on
the right hand side, you got 1.
244
00:15:00,64 --> 00:15:03,54
Slope 1 at x = 0.
245
00:15:03,54 --> 00:15:05,58
So that's great.
246
00:15:05,58 --> 00:15:08,21
Except of course, since we
don't know what e is, this
247
00:15:08,21 --> 00:15:15,77
is a little bit dicey.
248
00:15:15,77 --> 00:15:21,75
So, next even before
explaining what e is...
249
00:15:21,75 --> 00:15:24,02
In fact, we won't get to
what e really is until the
250
00:15:24,02 --> 00:15:26,11
very end of this lecture.
251
00:15:26,11 --> 00:15:34,53
But I have to persuade
you why e exists.
252
00:15:34,53 --> 00:15:38,3
We have to have some
explanation for why we know
253
00:15:38,3 --> 00:15:40,74
there is such a number.
254
00:15:40,74 --> 00:15:44,1
Ok, so first of all, let me
start with the one that we
255
00:15:44,1 --> 00:15:46,97
supposedly know, which
is the function 2^x .
256
00:15:46,97 --> 00:15:49,71
We'll call it f(x) is 2^x.
257
00:15:49,71 --> 00:15:50,46
All right?
258
00:15:50,46 --> 00:15:51,82
So that's the first thing.
259
00:15:51,82 --> 00:15:54,79
And remember, that the property
that it had, was that
260
00:15:54,79 --> 00:15:58,17
f '(0) was M(2) .
261
00:15:58,17 --> 00:16:01,44
That was the derivative of
this function, the slope
262
00:16:01,44 --> 00:16:06,61
of x = 0. of the graph.
263
00:16:06,61 --> 00:16:09,87
Of the tangent line, that is.
264
00:16:09,87 --> 00:16:14,53
So now, what we're going
to consider is any
265
00:16:14,53 --> 00:16:16,88
kind of stretching.
266
00:16:16,88 --> 00:16:22,75
We're going to stretch this
function by a factor k.
267
00:16:22,75 --> 00:16:23,61
Any number k.
268
00:16:23,61 --> 00:16:29,09
So what we're going to
consider is f(kx).
269
00:16:29,09 --> 00:16:34,79
If you do that, that's
the same as 2^kx.
270
00:16:34,79 --> 00:16:37,42
Right?
271
00:16:37,42 --> 00:16:41,03
But now if I use the second law
of exponents that I have over
272
00:16:41,03 --> 00:16:46,7
there, that's the same thing as
2 to the k to the power x,
273
00:16:46,7 --> 00:16:51,43
which is the same thing as some
base b ^ x, where
274
00:16:51,43 --> 00:16:54,26
b is equal to...
275
00:16:54,26 --> 00:16:59,3
Let's write that down
over here. b is 2^k .
276
00:16:59,3 --> 00:16:59,58
Right.
277
00:16:59,58 --> 00:17:03,63
So whatever it is, if I have
a different base which is
278
00:17:03,63 --> 00:17:08,32
expressed in terms of 2, by
being of the form 2^k , then
279
00:17:08,32 --> 00:17:14,11
that new function is
described by this function f
280
00:17:14,11 --> 00:17:17,7
(kx) , the stretch.
281
00:17:17,7 --> 00:17:20,73
So what happens when you
stretch a function?
282
00:17:20,73 --> 00:17:24,72
That's the same thing as
shrinking the x axis.
283
00:17:24,72 --> 00:17:30,11
So when k gets larger, this
corresponding point over here
284
00:17:30,11 --> 00:17:32,16
would be over here, and so
this corresponding point
285
00:17:32,16 --> 00:17:32,99
would be over here.
286
00:17:32,99 --> 00:17:38,91
So you shrink this picture,
and the slope here tilts up.
287
00:17:38,91 --> 00:17:43,14
So, as we increase k, the slope
gets steeper and steeper.
288
00:17:43,14 --> 00:17:47,57
Let's see that explicitly,
numerically here.
289
00:17:47,57 --> 00:17:51,87
Explicitly, numerically, if I
take the derivative here...
290
00:17:51,87 --> 00:17:59,34
So the derivative with
respect to x of b^ x, that's
291
00:17:59,34 --> 00:18:00,98
the chain rule, right?
292
00:18:00,98 --> 00:18:03,34
That's the derivative
with respect to x of
293
00:18:03,34 --> 00:18:08,26
f(kx), which is what?
294
00:18:08,26 --> 00:18:11,78
It's k times f '(kx) .
295
00:18:11,78 --> 00:18:23
And so if we do it at 0, we're
just getting k times f '(0) ,
296
00:18:23 --> 00:18:26,57
which is k times this M(2).
297
00:18:26,57 --> 00:18:31,39
So how is it exactly that we
cook up the right base b?
298
00:18:31,39 --> 00:18:40,26
So b = e when k = 1
over this number.
299
00:18:40,26 --> 00:18:44,65
In other words, we can pick all
possible slopes that we want.
300
00:18:44,65 --> 00:18:46,22
This just has the effect
of multiplying the
301
00:18:46,22 --> 00:18:47,55
slope by a factor.
302
00:18:47,55 --> 00:18:50,33
And we can shift the slope at
0 however we want, and we're
303
00:18:50,33 --> 00:18:56,63
going to do it so that the
slope exactly matches 1,
304
00:18:56,63 --> 00:18:58,15
the one that we want.
305
00:18:58,15 --> 00:18:59,58
We still don't know what k is.
306
00:18:59,58 --> 00:19:01,34
We still don't know what e is.
307
00:19:01,34 --> 00:19:04,94
But at least we know that
it's there somewhere.
308
00:19:04,94 --> 00:19:05,64
Yes?
309
00:19:05,64 --> 00:19:08,6
Student: How do you
know it's f(kx)?
310
00:19:08,6 --> 00:19:09,34
PROFESSOR: How do I know?
311
00:19:09,34 --> 00:19:13,44
Well, f(x) is 2 ^ x.
312
00:19:13,44 --> 00:19:19,16
If f(x) is 2^x, then the
formula for f(kx) is this.
313
00:19:19,16 --> 00:19:23,06
I've decided what f(x)
is, so therefore there's
314
00:19:23,06 --> 00:19:25,12
a formula for f(kx).
315
00:19:25,12 --> 00:19:27,31
And furthermore, by the chain
rule, there's a formula
316
00:19:27,31 --> 00:19:28,15
for the derivative.
317
00:19:28,15 --> 00:19:33,93
And it's k times the
derivative of f.
318
00:19:33,93 --> 00:19:35,15
So again, scaling does this.
319
00:19:35,15 --> 00:19:38,19
By the way, we did exactly
the same thing with the
320
00:19:38,19 --> 00:19:39,92
sine and cosine function.
321
00:19:39,92 --> 00:19:42,92
If you think of the sine
function here, let me just
322
00:19:42,92 --> 00:19:46,69
remind you here, what happens
with the chain rule, you get
323
00:19:46,69 --> 00:19:51,72
k times cosine k t here.
324
00:19:51,72 --> 00:19:55,34
So the fact that we set things
up beautifully with radians
325
00:19:55,34 --> 00:19:58,02
that this thing is, but we
could change the scale to
326
00:19:58,02 --> 00:20:02,34
anything, such as degrees, by
the appropriate factor k.
327
00:20:02,34 --> 00:20:05,55
And then there would be
this scale factor shift of
328
00:20:05,55 --> 00:20:07,54
the derivative formulas.
329
00:20:07,54 --> 00:20:09,72
Of course, the one with radians
is the easy one, because
330
00:20:09,72 --> 00:20:11,13
the factor is 1.
331
00:20:11,13 --> 00:20:14,74
The one with degrees is
horrible, because the factor is
332
00:20:14,74 --> 00:20:22,41
some crazy number like 180 over
pi, or something like that.
333
00:20:22,41 --> 00:20:26,21
Okay, so there's something
going on here which is
334
00:20:26,21 --> 00:20:30,42
exactly the same as that
kind of re-scaling.
335
00:20:30,42 --> 00:20:37,04
So, so far we've got only one
formula which is a keeper here.
336
00:20:37,04 --> 00:20:38,81
This one.
337
00:20:38,81 --> 00:20:41,12
We have a preliminary formula
that we still haven't
338
00:20:41,12 --> 00:20:45,82
completely explained which has
a little wavy line there.
339
00:20:45,82 --> 00:20:49,26
And we have to fit all
these things together.
340
00:20:49,26 --> 00:20:52,71
Okay, so now to fit them
together, I need to
341
00:20:52,71 --> 00:21:11,45
introduce the natural log.
342
00:21:11,45 --> 00:21:21,59
So the natural log is
denoted this way, ln(x).
343
00:21:21,59 --> 00:21:24,67
So maybe I'll call it a
new letter name, we'll
344
00:21:24,67 --> 00:21:28,47
call it w = ln x here.
345
00:21:28,47 --> 00:21:32,04
But if we were reversing
things, if we started out with
346
00:21:32,04 --> 00:21:37,88
a function y = e^x , the
property that it would have is
347
00:21:37,88 --> 00:21:41,03
that it's the inverse
function of e^x .
348
00:21:41,03 --> 00:21:46,17
So it has the property
that the lny = x.
349
00:21:46,17 --> 00:21:46,37
Right?
350
00:21:46,37 --> 00:21:58,91
So this defines the log.
351
00:21:58,91 --> 00:22:02,01
Now the logarithm has a bunch
of properties and they
352
00:22:02,01 --> 00:22:04,94
come from the exponential
properties in principle.
353
00:22:04,94 --> 00:22:07,5
You remember these.
354
00:22:07,5 --> 00:22:10,58
And I'm just going to
remind you of them.
355
00:22:10,58 --> 00:22:12,67
So the main one that I just
want to remind you of is that
356
00:22:12,67 --> 00:22:24,5
the ln (X1*X2) = ln X1
357
00:22:24,5 --> 00:22:28,13
ln X2.
358
00:22:28,13 --> 00:22:32,17
And maybe a few more are
worth reminding you of.
359
00:22:32,17 --> 00:22:37,12
One is that the
logarithm of 1 is 0.
360
00:22:37,12 --> 00:22:43,31
A second is that the
logarithm of e is 1.
361
00:22:43,31 --> 00:22:43,84
Alright?
362
00:22:43,84 --> 00:22:47,16
So these correspond to the
inverse relationships here.
363
00:22:47,16 --> 00:22:51,17
If I plug in here,
x = 0 and x = 1.
364
00:22:51,17 --> 00:22:56,65
If I plug in x = 0 and x = 1, I
get the corresponding numbers
365
00:22:56,65 --> 00:23:04,03
here: y = 1 and y = e.
366
00:23:04,03 --> 00:23:09,72
And maybe it would be worth
it to plot the picture
367
00:23:09,72 --> 00:23:13,43
once to reinforce this.
368
00:23:13,43 --> 00:23:16,62
So here I'll put them
on the same chart.
369
00:23:16,62 --> 00:23:20,2
If you have here e
to the x over here.
370
00:23:20,2 --> 00:23:21,79
It looks like this.
371
00:23:21,79 --> 00:23:28,61
Then the logarithm which I'll
maybe put in a different color.
372
00:23:28,61 --> 00:23:32,7
So this crosses at this all
important point here, (0,1).
373
00:23:32,7 --> 00:23:35,14
And now in order to figure out
what the inverse function
374
00:23:35,14 --> 00:23:40,75
is, I have to take the flip
across the diagonal, x = y.
375
00:23:40,75 --> 00:23:44,6
So that's this shape here,
going down like this.
376
00:23:44,6 --> 00:23:47,09
And here's the point (1, 0).
377
00:23:47,09 --> 00:23:50,7
So (1, 0) corresponds
to this identity here.
378
00:23:50,7 --> 00:23:53
But the log of 1 is 0.
379
00:23:53 --> 00:24:00,12
And notice, so this is ln
x, the graph of ln x.
380
00:24:00,12 --> 00:24:05,98
And notice it's only defined
for x positive, which
381
00:24:05,98 --> 00:24:09,57
corresponds to the fact that e
to the x is always positive.
382
00:24:09,57 --> 00:24:15,13
So in other words, this white
curve is only above this axis,
383
00:24:15,13 --> 00:24:19,21
and the orange one is
to the right here.
384
00:24:19,21 --> 00:24:27,99
It's only defined
for x positive.
385
00:24:27,99 --> 00:24:31,74
Oh, one other thing I should
mention is the slope here is 1.
386
00:24:31,74 --> 00:24:35,38
And so the slope there
is also going to be 1.
387
00:24:35,38 --> 00:24:41,18
Now, what we're allowed to do
relatively easily, because we
388
00:24:41,18 --> 00:24:44,47
have the tools to do it, is to
compute the derivative
389
00:24:44,47 --> 00:24:49,96
of the logarithm.
390
00:24:49,96 --> 00:25:01,74
So to find the derivative of
a log, we're going to use
391
00:25:01,74 --> 00:25:04,06
implicit differentiation.
392
00:25:04,06 --> 00:25:08,78
This is how we find
the derivative of any
393
00:25:08,78 --> 00:25:09,89
inverse function.
394
00:25:09,89 --> 00:25:12,13
So remember the way that works
is if you know the derivative
395
00:25:12,13 --> 00:25:14,02
of the function, you can find
the derivative of the
396
00:25:14,02 --> 00:25:15,59
inverse function.
397
00:25:15,59 --> 00:25:18,72
And the mechanism is the
following: you write
398
00:25:18,72 --> 00:25:22,68
down here w = ln x.
399
00:25:22,68 --> 00:25:23,51
Here's the function.
400
00:25:23,51 --> 00:25:25,51
We're trying to find
the derivative of w.
401
00:25:25,51 --> 00:25:28,75
But now we don't know how to
differentiate this equation,
402
00:25:28,75 --> 00:25:32,28
but if we exponentiate it,
so that's the same
403
00:25:32,28 --> 00:25:42,66
thing as e^w = x.
404
00:25:42,66 --> 00:25:46,42
Because let's just
stick this in here.
405
00:25:46,42 --> 00:25:52,33
e^ln x = x.
406
00:25:52,33 --> 00:25:54,68
Now we can differentiate this.
407
00:25:54,68 --> 00:25:56,8
So let's do the
differentiation here.
408
00:25:56,8 --> 00:26:04,01
We have d/dx e ^ w = d
/ dx of x, which is 1.
409
00:26:04,01 --> 00:26:07,555
And then this, by the
chain rule, is d / dw
410
00:26:07,555 --> 00:26:11,45
of e^w times dw /dx.
411
00:26:11,45 --> 00:26:14,56
The product of
these two factors.
412
00:26:14,56 --> 00:26:15,58
That's equal to 1.
413
00:26:15,58 --> 00:26:21,54
And now this guy, the one
little guy that we actually
414
00:26:21,54 --> 00:26:27,98
know and can use,
that's this guy here.
415
00:26:27,98 --> 00:26:33,8
So this is e^w times
dw/ dx, which is 1.
416
00:26:33,8 --> 00:26:44,73
And so finally, dw/
dx = 1 / e^w .
417
00:26:44,73 --> 00:26:47,08
But what is that?
418
00:26:47,08 --> 00:26:48,25
It's x.
419
00:26:48,25 --> 00:26:50,74
So this is 1 / x.
420
00:26:50,74 --> 00:26:55,32
So what we discovered is, and
now I get to put another green
421
00:26:55,32 --> 00:27:01,87
guy around here, is that
this is equal to 1/x.
422
00:27:01,87 --> 00:27:16,71
So alright, now we have two
companion formulas here.
423
00:27:16,71 --> 00:27:20,21
The rate of change
of ln x is 1 / x.
424
00:27:20,21 --> 00:27:24,83
And the rate of change of
e^x is itself, is e^x.
425
00:27:24,83 --> 00:27:31,56
And it's time to return to the
problem that we were having a
426
00:27:31,56 --> 00:27:35,25
little bit of trouble with,
which is somewhat not explicit,
427
00:27:35,25 --> 00:27:37,69
which is this M(a) times x.
428
00:27:37,69 --> 00:27:41,9
We want to now differentiate
a to the x in general;
429
00:27:41,9 --> 00:27:44,09
not just e^x .
430
00:27:44,09 --> 00:27:50,7
So let's work that out, and I
want to explain it in a couple
431
00:27:50,7 --> 00:27:52,71
of ways, so you're want to have
to remember this, because
432
00:27:52,71 --> 00:27:55,53
I'm going to erase it.
433
00:27:55,53 --> 00:28:02,45
But what I'd like you to do is,
so now I want to teach you how
434
00:28:02,45 --> 00:28:17,53
to differentiate basically
any exponential.
435
00:28:17,53 --> 00:28:31,58
So now to differentiate
any exponential.
436
00:28:31,58 --> 00:28:38,4
There are two methods.
437
00:28:38,4 --> 00:28:39,44
They're practically
the same method.
438
00:28:39,44 --> 00:28:41,48
They have the same
amount of arithmetic.
439
00:28:41,48 --> 00:28:45,61
You'll see both of them, and
they're equally valuable.
440
00:28:45,61 --> 00:28:48,15
So we're going to
just describes them.
441
00:28:48,15 --> 00:28:52,26
Method one I'm going
to illustrate on
442
00:28:52,26 --> 00:28:55,94
the function a^x .
443
00:28:55,94 --> 00:29:00,47
So we're interested in
differentiating this thing,
444
00:29:00,47 --> 00:29:04,28
exactly this problem that I
still didn't solve yet.
445
00:29:04,28 --> 00:29:05,08
Ok?
446
00:29:05,08 --> 00:29:06,95
So here it is.
447
00:29:06,95 --> 00:29:08,08
And here's the procedure.
448
00:29:08,08 --> 00:29:16,77
The procedure is to write, so
the method is to use base
449
00:29:16,77 --> 00:29:20,35
e, or convert to base e.
450
00:29:20,35 --> 00:29:22,43
So how do you
convert to base e?
451
00:29:22,43 --> 00:29:27,66
Well, you write a^x
as e to some power.
452
00:29:27,66 --> 00:29:29,02
So what power is it?
453
00:29:29,02 --> 00:29:34,98
It's e to the power ln
a, to the power x.
454
00:29:34,98 --> 00:29:40,73
And that is just e ^ x ln a.
455
00:29:40,73 --> 00:29:44,87
So we've made our
conversion now to base e.
456
00:29:44,87 --> 00:29:46,81
The exponential of something.
457
00:29:46,81 --> 00:29:50,41
So now I'm going to carry
out the differentiation.
458
00:29:50,41 --> 00:29:59,27
So d / dx of a ^x = d
/ dx of e ^ x ln a.
459
00:29:59,27 --> 00:30:05,82
And now, this is a step which
causes great confusion
460
00:30:05,82 --> 00:30:06,87
when you first see it.
461
00:30:06,87 --> 00:30:10,92
And you must get used to it,
because it's easy, not hard.
462
00:30:10,92 --> 00:30:13,45
Okay?
463
00:30:13,45 --> 00:30:18,82
The rate of change of this
with respect to x is, let
464
00:30:18,82 --> 00:30:23,04
me do it by analogy here.
465
00:30:23,04 --> 00:30:27,52
Because say I had e ^ 3X and
I were differentiating it.
466
00:30:27,52 --> 00:30:32,07
The chain rule would say that
this is just 3, the rate of
467
00:30:32,07 --> 00:30:36,33
change of 3X with respect
to x times e ^ 3X.
468
00:30:36,33 --> 00:30:41,06
The rate of change of e to
the u with respect to u.
469
00:30:41,06 --> 00:30:43,5
So this is the
ordinary chain rule.
470
00:30:43,5 --> 00:30:48,05
And what we're doing up here is
exactly the same thing, because
471
00:30:48,05 --> 00:30:51,97
ln a, as frightening as it
looks, with all three letters
472
00:30:51,97 --> 00:30:54,69
there, is just a fixed number.
473
00:30:54,69 --> 00:30:55,86
It's not moving.
474
00:30:55,86 --> 00:30:57,17
It's a constant.
475
00:30:57,17 --> 00:31:01,08
So the constant just
accelerates the rate of change
476
00:31:01,08 --> 00:31:04,98
by that factor, which is what
the chain rule is doing.
477
00:31:04,98 --> 00:31:11,83
So this is equal to ln
a times e ^ x ln a.
478
00:31:11,83 --> 00:31:17,39
Same business here with
ln a replacing 3.
479
00:31:17,39 --> 00:31:20,08
So this is something you've got
to get used to in time for the
480
00:31:20,08 --> 00:31:21,99
exam, for instance, because
you're going to be doing
481
00:31:21,99 --> 00:31:25,36
a million of these.
482
00:31:25,36 --> 00:31:27,81
So do get used to it.
483
00:31:27,81 --> 00:31:29,23
So here's the formula.
484
00:31:29,23 --> 00:31:33,34
On the other hand, this
expression here was
485
00:31:33,34 --> 00:31:34,73
the same as a ^ x.
486
00:31:34,73 --> 00:31:38,3
So another way of writing this,
and I'll put this into a
487
00:31:38,3 --> 00:31:41,79
box, but actually I never
remember this particularly.
488
00:31:41,79 --> 00:31:47,74
I just re-derive it every
time, is that the derivative
489
00:31:47,74 --> 00:31:51,1
of a^x = (ln a) a^x .
490
00:31:51,1 --> 00:31:56,93
Now I'm going to get rid
of what's underneath it.
491
00:31:56,93 --> 00:32:01,97
So this is another formula.
492
00:32:01,97 --> 00:32:05,5
So there's the formula I've
essentially finished here.
493
00:32:05,5 --> 00:32:11,19
And notice, what is
the magic number?
494
00:32:11,19 --> 00:32:16,12
The magic number is
the natural log of a.
495
00:32:16,12 --> 00:32:16,88
That's what it was.
496
00:32:16,88 --> 00:32:18,74
We didn't know what
it was in advance.
497
00:32:18,74 --> 00:32:19,47
This is what it is.
498
00:32:19,47 --> 00:32:21,45
It's the natural log of a.
499
00:32:21,45 --> 00:32:27,31
Let me emphasize to you again,
something about what's going on
500
00:32:27,31 --> 00:32:34,51
here, which has to do
with scale change.
501
00:32:34,51 --> 00:32:42,63
So, for example, the derivative
with respect to x of
502
00:32:42,63 --> 00:32:47,29
2^x is (ln 2) 2 ^ x.
503
00:32:47,29 --> 00:32:50,09
The derivative with respect to
x, these are the two most
504
00:32:50,09 --> 00:32:53,82
obvious bases that you might
want to use, is log
505
00:32:53,82 --> 00:32:56,76
10 times 10^x .
506
00:32:56,76 --> 00:32:59,47
So one of the things that's
natural about the natural
507
00:32:59,47 --> 00:33:05,57
logarithm is that even if we
insisted that we must use base
508
00:33:05,57 --> 00:33:08,73
2, or that we must use base 10,
we'd still be stuck with
509
00:33:08,73 --> 00:33:11,36
natural logarithms.
510
00:33:11,36 --> 00:33:12,54
They come up naturally.
511
00:33:12,54 --> 00:33:15,77
They're the ones which are
independent of our human
512
00:33:15,77 --> 00:33:20,19
construct of base
2 and base 10.
513
00:33:20,19 --> 00:33:23,1
The natural logarithm is
the one that comes up
514
00:33:23,1 --> 00:33:25,36
without reference.
515
00:33:25,36 --> 00:33:27,48
And we'll be mentioning a
few other ways in which
516
00:33:27,48 --> 00:33:31,11
it's natural later.
517
00:33:31,11 --> 00:33:35,55
So I told you about this first
method, now I want to tell you
518
00:33:35,55 --> 00:33:41,73
about a second method here.
519
00:33:41,73 --> 00:34:05,7
So the second is called
logarithmic differentiation.
520
00:34:05,7 --> 00:34:07,77
So how does this work?
521
00:34:07,77 --> 00:34:14,83
Well, sometimes you're having
trouble differentiating a
522
00:34:14,83 --> 00:34:21,78
function, and it's easier to
differentiate its logarithm.
523
00:34:21,78 --> 00:34:24,31
That may seem peculiar, but
actually we'll give several
524
00:34:24,31 --> 00:34:27,47
examples where this is clearly
the case, that the logarithm is
525
00:34:27,47 --> 00:34:30,83
easier to differentiate
than the function.
526
00:34:30,83 --> 00:34:34,09
So it could be that this is an
easier quantity to understand.
527
00:34:34,09 --> 00:34:39,47
So we want to relate it
back to the function u.
528
00:34:39,47 --> 00:34:44,17
So I'm going to write it a
slightly different way.
529
00:34:44,17 --> 00:34:47,27
Let's write it in
terms of primes here.
530
00:34:47,27 --> 00:34:51,38
So the basic identity is the
chain rule again, and the
531
00:34:51,38 --> 00:34:53,39
derivative of the logarithm,
well maybe I'll write
532
00:34:53,39 --> 00:34:54,92
it out this way first.
533
00:34:54,92 --> 00:35:05,12
So this would be d of ln
u/ du, times du / dx .
534
00:35:05,12 --> 00:35:10,11
These are the two factors.
535
00:35:10,11 --> 00:35:12,66
And that's the same thing, so
remember what the derivative
536
00:35:12,66 --> 00:35:14,14
of the logarithm is.
537
00:35:14,14 --> 00:35:17,82
This is 1 / u.
538
00:35:17,82 --> 00:35:23,57
So here I have a 1 / u, and
here I have a du / dx.
539
00:35:23,57 --> 00:35:29,49
So I'm going to encode this on
the next board here, which is
540
00:35:29,49 --> 00:35:32,54
sort of the main formula you
always need to remember, which
541
00:35:32,54 --> 00:35:39,53
is that (ln u)' = u' / u.
542
00:35:39,53 --> 00:35:42,61
That's the one to
remember here.
543
00:35:42,61 --> 00:35:47,32
STUDENT: [INAUDIBLE].
544
00:35:47,32 --> 00:35:51,87
PROFESSOR: The question is how
did I get this step here?
545
00:35:51,87 --> 00:35:58,5
So this is the chain rule.
546
00:35:58,5 --> 00:36:02,47
The rate of change of ln u with
respect to x is the rate of
547
00:36:02,47 --> 00:36:05,496
change of ln u with respect u,
times the rate of change
548
00:36:05,496 --> 00:36:07,73
of u with respect to x.
549
00:36:07,73 --> 00:36:18,56
That's the chain rule.
550
00:36:18,56 --> 00:36:23,48
So now I've worked out this
identity here, and now let's
551
00:36:23,48 --> 00:36:30,73
show how it handles this
case, d / dx of a^x.
552
00:36:30,73 --> 00:36:31,74
Let's do this one.
553
00:36:31,74 --> 00:36:39,57
So in order to get that one,
I would take u = a^x .
554
00:36:39,57 --> 00:36:43,06
And now let's just take
a look at what ln u is.
555
00:36:43,06 --> 00:36:51,46
Ln u = x ln a.
556
00:36:51,46 --> 00:36:55,01
Now I claimed that this is
pretty easy to differentiate.
557
00:36:55,01 --> 00:37:00,2
Again, it may seem hard, but
it's actually quite easy.
558
00:37:00,2 --> 00:37:04,59
So maybe somebody
can hazard a guess.
559
00:37:04,59 --> 00:37:11,53
What's the derivative
of x ln a?
560
00:37:11,53 --> 00:37:14,87
It's just log a.
561
00:37:14,87 --> 00:37:18,07
So this is the same thing that
I was talking about before,
562
00:37:18,07 --> 00:37:22,18
which is if you've got 3X, and
you're taking its derivative
563
00:37:22,18 --> 00:37:25,02
with respect to x
here, that's just 3.
564
00:37:25,02 --> 00:37:26,22
That's the kind of
thing you have.
565
00:37:26,22 --> 00:37:30,11
Again, don't be put off by this
massive piece of junk here.
566
00:37:30,11 --> 00:37:33,26
It's a constant.
567
00:37:33,26 --> 00:37:38,22
So again, keep that in mind.
568
00:37:38,22 --> 00:37:42,46
It comes up regularly in
this kind of question.
569
00:37:42,46 --> 00:37:44,69
So there's is our formula,
that the logarithmic
570
00:37:44,69 --> 00:37:46,98
derivative is this.
571
00:37:46,98 --> 00:37:50,36
But let's just rewrite that.
572
00:37:50,36 --> 00:37:55,4
That's the same thing as u'
/ u, which is log u prime
573
00:37:55,4 --> 00:37:58,6
is equal to ln a, right?
574
00:37:58,6 --> 00:38:00,61
So this is our
differentiation formula.
575
00:38:00,61 --> 00:38:01,8
So here we have u'.
576
00:38:01,8 --> 00:38:07,46
u' is equal to u times ln a, if
I just multiply through by u.
577
00:38:07,46 --> 00:38:08,63
And that's what we wanted.
578
00:38:08,63 --> 00:38:16,66
That's d / dx of a^x = ln
a (I'll reverse the order
579
00:38:16,66 --> 00:38:24,64
of the two, which is
customary) times a^x.
580
00:38:24,64 --> 00:38:26,78
So this is the way
that logarithmic
581
00:38:26,78 --> 00:38:27,89
differentiation works.
582
00:38:27,89 --> 00:38:33,41
It's the same arithmetic as the
previous method, but we don't
583
00:38:33,41 --> 00:38:34,93
have to convert to base e.
584
00:38:34,93 --> 00:38:38,37
We're just keeping track of
the exponents and doing
585
00:38:38,37 --> 00:38:40,26
differentiation on the
exponents, and multiplying
586
00:38:40,26 --> 00:38:44,33
through at the end.
587
00:38:44,33 --> 00:38:49,66
Okay, so I'm going to do two
trickier examples, which
588
00:38:49,66 --> 00:39:02,4
illustrate logarithmic
differentiation.
589
00:39:02,4 --> 00:39:05,83
Again, these could be done
equally well by using base
590
00:39:05,83 --> 00:39:07,73
e, but I won't do it.
591
00:39:07,73 --> 00:39:12,12
Method one and method
two always both work.
592
00:39:12,12 --> 00:39:16,73
So here's a second example:
again this is a problem when
593
00:39:16,73 --> 00:39:23,22
you have moving exponents.
594
00:39:23,22 --> 00:39:26,35
But this time, we're going to
complicate matters by having
595
00:39:26,35 --> 00:39:30,52
both a moving exponent
and a moving base.
596
00:39:30,52 --> 00:39:34,19
So we have a function u, which
is, well maybe I'll call it
597
00:39:34,19 --> 00:39:38,64
v, since we already had a
function u, which is x ^ x.
598
00:39:38,64 --> 00:39:41,67
A really complicated
looking function here.
599
00:39:41,67 --> 00:39:44,665
So again you can handle
this by converting to
600
00:39:44,665 --> 00:39:47,22
base e, method one.
601
00:39:47,22 --> 00:39:49,18
But we'll do the logarithmic
differentiation
602
00:39:49,18 --> 00:39:51,11
version, alright?
603
00:39:51,11 --> 00:39:59,31
So I take the logs
of both sides.
604
00:39:59,31 --> 00:40:04,37
And now I differentiate it.
605
00:40:04,37 --> 00:40:06,5
And now when I differentiate
this here, I have to
606
00:40:06,5 --> 00:40:07,73
use the product rule.
607
00:40:07,73 --> 00:40:09,57
This time, instead of
having ln a, a constant,
608
00:40:09,57 --> 00:40:10,98
I have a variable here.
609
00:40:10,98 --> 00:40:12,66
So I have two factors.
610
00:40:12,66 --> 00:40:14,84
I have ln x when I
differentiate with
611
00:40:14,84 --> 00:40:15,416
respect to x.
612
00:40:15,416 --> 00:40:19,5
When I differentiate with
respect to this factor here, I
613
00:40:19,5 --> 00:40:26,91
get that x times the derivative
of that, which is 1/x.
614
00:40:26,91 --> 00:40:29,16
So, here's my formula.
615
00:40:29,16 --> 00:40:30,43
Almost finished.
616
00:40:30,43 --> 00:40:33,09
So I have here v' / v.
617
00:40:33,09 --> 00:40:35,2
I'm going to multiply these
two things together.
618
00:40:35,2 --> 00:40:37,38
I'll put it on the other side,
because I don't want to get it
619
00:40:37,38 --> 00:40:45,34
mixed up with ln x plus
one, the quantity.
620
00:40:45,34 --> 00:40:47,1
And now I'm almost done.
621
00:40:47,1 --> 00:40:53,8
I have v' = v (1
622
00:40:53,8 --> 00:41:04,31
ln x), and that's just d /
dx( x ^x) = x^x (1 + lnx).
623
00:41:04,31 --> 00:41:13,45
That's it.
624
00:41:13,45 --> 00:41:32,1
So these two methods always
work for moving exponents.
625
00:41:32,1 --> 00:41:34,32
So the next thing that I'd
like to do is another
626
00:41:34,32 --> 00:41:36,22
fairly tricky example.
627
00:41:36,22 --> 00:41:45,88
And this one is not strictly
speaking within Calculus.
628
00:41:45,88 --> 00:41:50,39
Although we're going to use the
tools that we just described to
629
00:41:50,39 --> 00:41:52,82
carry it out, in fact it will
use some Calculus
630
00:41:52,82 --> 00:41:55,99
in the very end.
631
00:41:55,99 --> 00:41:59,75
And what I'm going to do is I'm
going to evaluate the limit as
632
00:41:59,75 --> 00:42:11,43
n goes to infinity of one plus
one over n to the power n.
633
00:42:11,43 --> 00:42:16,17
So now, the reason why I want
to discuss this is, is it
634
00:42:16,17 --> 00:42:18,46
turns out to have a very
interesting answer.
635
00:42:18,46 --> 00:42:22,94
And it's a problem
that you can approach
636
00:42:22,94 --> 00:42:24,49
exactly by this method.
637
00:42:24,49 --> 00:42:28,58
And the reason is that it
has a moving exponent.
638
00:42:28,58 --> 00:42:30,84
The exponent n
here is changing.
639
00:42:30,84 --> 00:42:33,1
And so if you want to keep
track of that, a good way to
640
00:42:33,1 --> 00:42:36,74
do that is to use logarithms.
641
00:42:36,74 --> 00:42:39,37
So in order to figure out this
limit, we're going to take the
642
00:42:39,37 --> 00:42:42,16
log of it and figure out what
the limit of the log is,
643
00:42:42,16 --> 00:42:43,28
instead of the log
of the limit.
644
00:42:43,28 --> 00:42:44,99
Those will be the same thing.
645
00:42:44,99 --> 00:42:48,85
So we're going to take the
natural log of this quantity
646
00:42:48,85 --> 00:42:55,95
here, and that's n ln (1
647
00:42:55,95 --> 00:43:02,64
(1 / n)).
648
00:43:02,64 --> 00:43:07,43
And now I'm going to rewrite
this in a form which will make
649
00:43:07,43 --> 00:43:17,62
it more recognizable, so what
I'd like to do is I'm going to
650
00:43:17,62 --> 00:43:24,73
write n or maybe I should say
it this way: delta x = 1 / n.
651
00:43:24,73 --> 00:43:30,07
So if n is going to infinity,
then this delta x is
652
00:43:30,07 --> 00:43:33,7
going to be going to 0.
653
00:43:33,7 --> 00:43:36,85
So this is more familiar
territory for us in
654
00:43:36,85 --> 00:43:38,56
this class, anyway.
655
00:43:38,56 --> 00:43:40,37
So let's rewrite it.
656
00:43:40,37 --> 00:43:42,98
So here, we have
one over delta x.
657
00:43:42,98 --> 00:43:48,03
And then that is
multiplied by ln 1
658
00:43:48,03 --> 00:43:50,15
delta x.
659
00:43:50,15 --> 00:43:54,86
So n is the reciprocal
of delta x.
660
00:43:54,86 --> 00:43:58,33
Now I want to change this
in a very, very minor way.
661
00:43:58,33 --> 00:44:01,15
I'm going to
subtract 0 from it.
662
00:44:01,15 --> 00:44:02,47
So that's the same thing.
663
00:44:02,47 --> 00:44:06,42
So what I'm going to do is I'm
going to subtract ln 1 from it.
664
00:44:06,42 --> 00:44:08,26
That's just equal to 0.
665
00:44:08,26 --> 00:44:10,55
So this is not a problem,
and I'll put some
666
00:44:10,55 --> 00:44:14,8
parentheses around it.
667
00:44:14,8 --> 00:44:18,08
Now you're supposed to
recognize, all of a sudden,
668
00:44:18,08 --> 00:44:21,28
what pattern this fits into.
669
00:44:21,28 --> 00:44:25,65
This is the thing which we need
to calculate in order to
670
00:44:25,65 --> 00:44:30,81
calculate the derivative
of the log function.
671
00:44:30,81 --> 00:44:36,57
So this is in the limit as
delta x goes to 0 equal to
672
00:44:36,57 --> 00:44:39,3
the derivative of ln x.
673
00:44:39,3 --> 00:44:39,89
Where?
674
00:44:39,89 --> 00:44:43,94
Well the base point is x=1.
675
00:44:43,94 --> 00:44:45,71
That's where we're
evaluating it.
676
00:44:45,71 --> 00:44:46,73
That's the X0.
677
00:44:46,73 --> 00:44:49,38
That's the base value.
678
00:44:49,38 --> 00:44:51
So this is the
difference quotient.
679
00:44:51 --> 00:44:52,43
That's exactly what it is.
680
00:44:52,43 --> 00:44:57,63
And so this by definition
tends to the limit here.
681
00:44:57,63 --> 00:45:01,47
But we know what the derivative
of the log function is.
682
00:45:01,47 --> 00:45:09,11
The derivative of the
log function is 1 / x.
683
00:45:09,11 --> 00:45:17,47
So this limit is 1.
684
00:45:17,47 --> 00:45:18,5
So we got it.
685
00:45:18,5 --> 00:45:19,73
We got the limit.
686
00:45:19,73 --> 00:45:23,47
And now we just have to work
backwards to figure out what
687
00:45:23,47 --> 00:45:34,01
this limit that we've
got over here is.
688
00:45:34,01 --> 00:45:37,13
So let's do that.
689
00:45:37,13 --> 00:45:40,57
So let's see here the
log approached 1.
690
00:45:40,57 --> 00:45:46,74
So the limit as n goes
to infinity of 1 plus
691
00:45:46,74 --> 00:45:49,53
(1 over n) to the n.
692
00:45:49,53 --> 00:45:51,88
So sorry, the log of this.
693
00:45:51,88 --> 00:45:54,32
Yeah, let's write it this way.
694
00:45:54,32 --> 00:45:57,53
It's the same thing, as well,
the thing that we know
695
00:45:57,53 --> 00:46:00,61
is the log of this.
696
00:46:00,61 --> 00:46:04,7
1 plus 1 over n to the n.
697
00:46:04,7 --> 00:46:06,39
And goes to infinity.
698
00:46:06,39 --> 00:46:08,19
That's the one that
we just figured out.
699
00:46:08,19 --> 00:46:11,38
But now this thing is the
exponential of that.
700
00:46:11,38 --> 00:46:16,55
So it's really e to
this power here.
701
00:46:16,55 --> 00:46:20,146
So this guy is the same as the
limit of the log of the limit
702
00:46:20,146 --> 00:46:22,34
of the thing, which is the
same as log of the limit.
703
00:46:22,34 --> 00:46:32,26
The limit of the log and the
log of the limit are the same.
704
00:46:32,26 --> 00:46:34,28
Ok, so I take the logarithm,
then I'm going to
705
00:46:34,28 --> 00:46:35,17
take the exponential.
706
00:46:35,17 --> 00:46:37,8
That just undoes
what I did before.
707
00:46:37,8 --> 00:46:41,91
And so this limit is
just 1, is e ^ 1.
708
00:46:41,91 --> 00:46:52
And so the limit that we
want here is equal to e.
709
00:46:52 --> 00:46:56,23
So I claim that with this
step, we've actually
710
00:46:56,23 --> 00:46:58,27
closed the loop, finally.
711
00:46:58,27 --> 00:47:03,62
Because we have an honest
numerical way to calculate e.
712
00:47:03,62 --> 00:47:04,14
The first.
713
00:47:04,14 --> 00:47:05,4
There are many such.
714
00:47:05,4 --> 00:47:07,51
But this one is a perfectly
honest numerical
715
00:47:07,51 --> 00:47:08,68
way to calculate e.
716
00:47:08,68 --> 00:47:09,95
We had this thing.
717
00:47:09,95 --> 00:47:12,17
We didn't know
exactly what it was.
718
00:47:12,17 --> 00:47:14,9
It was this M(e), there was
M(a), the logarithm, and so on.
719
00:47:14,9 --> 00:47:15,83
We have all that stuff.
720
00:47:15,83 --> 00:47:17,66
But we really didn't
need to nail down what
721
00:47:17,66 --> 00:47:18,92
this number e is.
722
00:47:18,92 --> 00:47:22,59
And this is telling us, if you
take for example 1 plus 1 over
723
00:47:22,59 --> 00:47:27,11
100 to the 100th power, that's
going to be a very good,
724
00:47:27,11 --> 00:47:30,73
perfectly decent anyway,
approximation to e.
725
00:47:30,73 --> 00:47:36,97
So this is a numerical
approximation, which is all we
726
00:47:36,97 --> 00:47:42,7
can ever do with just this
kind of irrational number.
727
00:47:42,7 --> 00:47:48,5
And so that closes the loop,
and we now have a coherent
728
00:47:48,5 --> 00:47:51,355
family of functions, which are
actually well defined and for
729
00:47:51,355 --> 00:47:54,55
which we have practical
methods to calculate.
730
00:47:54,55 --> 00:47:56,8
Okay see you next time.
731
00:47:56,8 --> 00:47:56,812