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PROFESSOR: All right, so
let's begin lecture six.
9
00:00:27,060 --> 00:00:44,630
We're talking today about
exponentials and logarithms.
10
00:00:44,630 --> 00:00:49,390
And these are the last functions
that I need to introduce,
11
00:00:49,390 --> 00:00:50,950
the last standard
functions that we
12
00:00:50,950 --> 00:00:55,110
need to connect with calculus,
that you've learned about.
13
00:00:55,110 --> 00:00:58,230
And they're certainly as
fundamental, if not more so,
14
00:00:58,230 --> 00:01:00,830
than trigonometric functions.
15
00:01:00,830 --> 00:01:04,960
So first of all, we'll
start out with a number,
16
00:01:04,960 --> 00:01:09,030
a, which is positive, which
is usually called a base.
17
00:01:09,030 --> 00:01:12,790
And then we have
these properties that
18
00:01:12,790 --> 00:01:14,870
a to the power 0 is always 1.
19
00:01:14,870 --> 00:01:17,180
That's how we get started.
20
00:01:17,180 --> 00:01:21,270
And a^1 is a.
21
00:01:21,270 --> 00:01:24,030
And of course a^2,
not surprisingly,
22
00:01:24,030 --> 00:01:26,200
is a times a, etc.
23
00:01:26,200 --> 00:01:35,400
And the general rule is that
a^(x_1 + x_2) is a^(x_1) times
24
00:01:35,400 --> 00:01:36,970
a^(x_2).
25
00:01:36,970 --> 00:01:41,440
So this is the basic rule of
exponents, and with these two
26
00:01:41,440 --> 00:01:49,070
initial properties, that defines
the exponential function.
27
00:01:49,070 --> 00:01:52,220
And then there's an
additional property,
28
00:01:52,220 --> 00:01:54,360
which is deduced
from these, which
29
00:01:54,360 --> 00:01:59,020
is the composition of
exponential functions, which
30
00:01:59,020 --> 00:02:03,330
is that you take a to the
x_1 power, to the x_2 power.
31
00:02:03,330 --> 00:02:08,390
Then that turns out to be
a to the x_1 times x_2.
32
00:02:08,390 --> 00:02:11,540
So that's an additional property
that we'll take for granted,
33
00:02:11,540 --> 00:02:14,140
which you learned
in high school.
34
00:02:14,140 --> 00:02:22,650
Now, in order to understand
what all the values of a^x are,
35
00:02:22,650 --> 00:02:28,620
we need to first remember that
if you're taking a rational
36
00:02:28,620 --> 00:02:34,350
power that it's the ratio
of two integers power of a.
37
00:02:34,350 --> 00:02:37,190
That's going to be a^m, and
then we're going to have to take
38
00:02:37,190 --> 00:02:39,440
the nth root of that.
39
00:02:39,440 --> 00:02:40,840
So that's the definition.
40
00:02:40,840 --> 00:02:45,050
And then, when
you're defining a^x,
41
00:02:45,050 --> 00:03:00,370
so a^x is defined for
all x by filling in.
42
00:03:00,370 --> 00:03:03,480
So I'm gonna use that
expression in quotation marks,
43
00:03:03,480 --> 00:03:09,930
"filling in" by continuity.
44
00:03:09,930 --> 00:03:11,540
This is really what
your calculator
45
00:03:11,540 --> 00:03:13,960
does when it gives
you a to the power x,
46
00:03:13,960 --> 00:03:17,540
because you can't even punch
in the square root of x.
47
00:03:17,540 --> 00:03:19,590
It doesn't really exist
on your calculator.
48
00:03:19,590 --> 00:03:21,230
There's some decimal expansion.
49
00:03:21,230 --> 00:03:24,400
So it takes the decimal
expansion to a certain length
50
00:03:24,400 --> 00:03:26,080
and spits out a
number which is pretty
51
00:03:26,080 --> 00:03:28,120
close to the correct answer.
52
00:03:28,120 --> 00:03:31,550
But indeed, in theory,
there is an a to the power
53
00:03:31,550 --> 00:03:33,710
square root of 2, even
though the square root of 2
54
00:03:33,710 --> 00:03:34,740
is irrational.
55
00:03:34,740 --> 00:03:37,540
And there's a to
the pi and so forth.
56
00:03:37,540 --> 00:03:41,000
All right, so that's the
exponential function,
57
00:03:41,000 --> 00:03:46,830
and let's draw a picture of one.
58
00:03:46,830 --> 00:03:52,230
So we'll try, say y = 2^x here.
59
00:03:52,230 --> 00:03:55,180
And I'm not going to draw
such a careful graph,
60
00:03:55,180 --> 00:03:58,630
but let's just plot the
most important point, which
61
00:03:58,630 --> 00:04:01,360
is the point (0,1).
62
00:04:01,360 --> 00:04:04,510
That's 2^0, which is 1.
63
00:04:04,510 --> 00:04:08,940
And then maybe we'll go
back up here to -1 here.
64
00:04:08,940 --> 00:04:13,860
And 2 to the -1 is
this point here.
65
00:04:13,860 --> 00:04:18,990
This is (-1, 1/2),
the reciprocal.
66
00:04:18,990 --> 00:04:23,460
And over here, we have 1, and so
that goes all the way up to 2.
67
00:04:23,460 --> 00:04:26,870
And then exponentials
are remarkably fast.
68
00:04:26,870 --> 00:04:30,680
So it's off the board what
happens next out at 2.
69
00:04:30,680 --> 00:04:34,360
It's already above
my range here,
70
00:04:34,360 --> 00:04:37,970
but the graph looks
something like this.
71
00:04:37,970 --> 00:04:38,610
All right.
72
00:04:38,610 --> 00:04:41,200
Now I've just visually,
at least, graphically
73
00:04:41,200 --> 00:04:43,240
filled in all the
rest of the points.
74
00:04:43,240 --> 00:04:46,730
You have to imagine all these
rational numbers, and so forth.
75
00:04:46,730 --> 00:04:51,670
So this point here
would have been (1, 2).
76
00:04:51,670 --> 00:04:53,330
And so forth.
77
00:04:53,330 --> 00:04:54,610
All right?
78
00:04:54,610 --> 00:05:00,992
So that's not too far along.
79
00:05:00,992 --> 00:05:01,950
So now what's our goal?
80
00:05:01,950 --> 00:05:04,020
Well, obviously we want
to do calculus here.
81
00:05:04,020 --> 00:05:08,050
So our goal, here, for now -
and it's gonna take a while.
82
00:05:08,050 --> 00:05:10,220
We have to think
about it pretty hard.
83
00:05:10,220 --> 00:05:22,250
We have to calculate
what this derivative is.
84
00:05:22,250 --> 00:05:26,020
All right, so we'll get started.
85
00:05:26,020 --> 00:05:28,520
And the way we get
started is simply
86
00:05:28,520 --> 00:05:31,190
by plugging in the
definition of the derivative.
87
00:05:31,190 --> 00:05:34,760
The derivative is
the limit as delta
88
00:05:34,760 --> 00:05:41,690
x goes to 0 of a to the x plus
delta x, minus a to the x,
89
00:05:41,690 --> 00:05:45,080
divided by delta x.
90
00:05:45,080 --> 00:05:50,320
So that's what it is.
91
00:05:50,320 --> 00:05:56,140
And now, the only step that
we can really perform here
92
00:05:56,140 --> 00:05:58,790
to make this is into
something a little bit simpler
93
00:05:58,790 --> 00:06:03,200
is to use this very first
rule that we have here.
94
00:06:03,200 --> 00:06:06,930
That the exponential of
the sum is the product
95
00:06:06,930 --> 00:06:08,070
of the exponentials.
96
00:06:08,070 --> 00:06:10,340
So we have here, a^x .
97
00:06:10,340 --> 00:06:15,750
So what I want to use is just
the property that a^(x + delta
98
00:06:15,750 --> 00:06:22,460
x) = a^x a^(delta x).
99
00:06:22,460 --> 00:06:26,880
And if I do that, I see that I
can factor out a common factor
100
00:06:26,880 --> 00:06:29,760
in the numerator, which is a^x.
101
00:06:29,760 --> 00:06:35,120
So we'll write this as the
limit as delta x goes to 0,
102
00:06:35,120 --> 00:06:41,300
of a to the x times this ratio,
now a to the delta x, minus 1,
103
00:06:41,300 --> 00:06:49,280
divided by delta x.
104
00:06:49,280 --> 00:06:50,160
So far, so good?
105
00:06:50,160 --> 00:06:53,640
We're actually almost to
some serious progress here.
106
00:06:53,640 --> 00:06:58,460
So there's one other
important conceptual step
107
00:06:58,460 --> 00:07:00,100
which we need to understand.
108
00:07:00,100 --> 00:07:03,010
And this is a
relatively simple one.
109
00:07:03,010 --> 00:07:05,340
We actually did this
before, by the way.
110
00:07:05,340 --> 00:07:08,162
We did this with
sines and cosines.
111
00:07:08,162 --> 00:07:09,870
The next thing I want
to point out to you
112
00:07:09,870 --> 00:07:15,680
is that you're used to thinking
of x as being the variable.
113
00:07:15,680 --> 00:07:18,140
And indeed, already
we were discussing
114
00:07:18,140 --> 00:07:20,130
x as being the variable
and a as being fixed.
115
00:07:20,130 --> 00:07:22,220
But for the purposes
of this limit,
116
00:07:22,220 --> 00:07:26,290
there's a different variable
that's moving. x is fixed
117
00:07:26,290 --> 00:07:29,340
and delta x is the
thing that's moving.
118
00:07:29,340 --> 00:07:33,440
So that means that this factor
here, which is a common factor,
119
00:07:33,440 --> 00:07:34,799
is constant.
120
00:07:34,799 --> 00:07:36,590
And we can just factor
it out of the limit.
121
00:07:36,590 --> 00:07:39,450
It doesn't affect
the limit at all.
122
00:07:39,450 --> 00:07:41,130
A constant times a
limit is the same
123
00:07:41,130 --> 00:07:44,580
as whether we multiply before
or after we take the limit.
124
00:07:44,580 --> 00:07:46,830
So I'm just going
to factor that out.
125
00:07:46,830 --> 00:07:49,110
So that's my next step here.
126
00:07:49,110 --> 00:07:53,450
a^x, and then I have the
limit delta x goes to 0
127
00:07:53,450 --> 00:07:59,830
of a to the delta x minus
1, divided by delta x.
128
00:07:59,830 --> 00:08:02,180
All right?
129
00:08:02,180 --> 00:08:03,910
And so what I have
here, so this is
130
00:08:03,910 --> 00:08:05,220
by definition the derivative.
131
00:08:05,220 --> 00:08:12,730
So here is d/dx of a^x, and it's
equal to this expression here.
132
00:08:12,730 --> 00:08:17,340
Now, I want to stare
at this expression,
133
00:08:17,340 --> 00:08:22,880
and see what it's telling
us, because it's telling us
134
00:08:22,880 --> 00:08:27,680
as much as we can get
so far, without some--
135
00:08:27,680 --> 00:08:34,810
So first let's just
look at what this says.
136
00:08:34,810 --> 00:08:40,260
So what it's saying is that the
derivative of a^x is a^x times
137
00:08:40,260 --> 00:08:42,820
something that we
don't yet know.
138
00:08:42,820 --> 00:08:46,060
And I'm going to call this
something, this mystery number,
139
00:08:46,060 --> 00:08:47,130
M(a).
140
00:08:47,130 --> 00:08:52,990
So I'm gonna make the label,
M(a) is equal to the limit
141
00:08:52,990 --> 00:08:56,640
as delta x goes to 0
of a to the delta x
142
00:08:56,640 --> 00:08:59,800
minus 1 divided by delta x.
143
00:08:59,800 --> 00:09:00,300
All right?
144
00:09:00,300 --> 00:09:08,870
So this is a definition.
145
00:09:08,870 --> 00:09:14,540
So this mystery number M(a)
has a geometric interpretation,
146
00:09:14,540 --> 00:09:16,000
as well.
147
00:09:16,000 --> 00:09:17,554
So let me describe that.
148
00:09:17,554 --> 00:09:18,970
It has a geometric
interpretation,
149
00:09:18,970 --> 00:09:20,678
and it's a very, very
significant number.
150
00:09:20,678 --> 00:09:22,200
So let's work out what that is.
151
00:09:22,200 --> 00:09:25,690
So first of all, let's rewrite
the expression in the box,
152
00:09:25,690 --> 00:09:28,470
using the shorthand
for this number.
153
00:09:28,470 --> 00:09:33,840
So if I just rewrite it, it says
d/dx of a^x is equal to this
154
00:09:33,840 --> 00:09:37,800
factor, which is
M(a), times a^x.
155
00:09:37,800 --> 00:09:43,450
So the derivative of the
exponential is this mystery
156
00:09:43,450 --> 00:09:44,790
number times a^x.
157
00:09:44,790 --> 00:09:48,870
So we've almost solved
the problem of finding
158
00:09:48,870 --> 00:09:50,560
the derivative of a^x.
159
00:09:50,560 --> 00:09:53,270
We just have to figure
out this one number, M(a),
160
00:09:53,270 --> 00:09:55,720
and we get the rest.
161
00:09:55,720 --> 00:10:01,780
So let me point out two more
things about this number, M(a).
162
00:10:01,780 --> 00:10:09,350
So first of all,
if I plug in x = 0,
163
00:10:09,350 --> 00:10:14,250
that's going to be
d/dx of a^x , at x = 0.
164
00:10:14,250 --> 00:10:19,150
According to this formula,
that's M(a) times a^0,
165
00:10:19,150 --> 00:10:21,370
which of course M(a).
166
00:10:21,370 --> 00:10:23,540
So what is M(a)?
167
00:10:23,540 --> 00:10:26,410
M(a) is the derivative
of this function at 0.
168
00:10:26,410 --> 00:10:39,790
So M(a) is the slope of
a^x at x = 0, of the graph.
169
00:10:39,790 --> 00:10:41,330
The graph of a^x at 0.
170
00:10:41,330 --> 00:10:46,170
So again over here, if
you looked at the picture.
171
00:10:46,170 --> 00:10:48,260
I'll draw the one
tangent line in here,
172
00:10:48,260 --> 00:10:50,640
which is this one here.
173
00:10:50,640 --> 00:11:00,050
And this thing has slope,
what we're calling M(2).
174
00:11:00,050 --> 00:11:02,491
So, if I graph the
function y = 2^x,
175
00:11:02,491 --> 00:11:03,740
I'll get a certain slope here.
176
00:11:03,740 --> 00:11:05,315
If I graph it with
a different base,
177
00:11:05,315 --> 00:11:07,590
I might get another slope.
178
00:11:07,590 --> 00:11:12,820
And what we got so far is
the following phenomenon:
179
00:11:12,820 --> 00:11:16,310
if we know this one number, if
we know the slope at this one
180
00:11:16,310 --> 00:11:18,940
place, we will be able to figure
out the formula for the slope
181
00:11:18,940 --> 00:11:23,320
everywhere else.
182
00:11:23,320 --> 00:11:25,690
Now, that's actually
exactly the same thing
183
00:11:25,690 --> 00:11:28,040
that we did for
sines and cosines.
184
00:11:28,040 --> 00:11:33,120
We knew the slope of the
sine and the cosine function
185
00:11:33,120 --> 00:11:35,900
at x = 0.
186
00:11:35,900 --> 00:11:37,450
The sine function had slope 1.
187
00:11:37,450 --> 00:11:39,470
The cosine function had slope 0.
188
00:11:39,470 --> 00:11:41,560
And then from the
sum formulas, well
189
00:11:41,560 --> 00:11:44,004
that's exactly this
kind of thing here,
190
00:11:44,004 --> 00:11:44,920
from the sum formulas.
191
00:11:44,920 --> 00:11:47,110
This sum formula, in fact
is easier than the ones
192
00:11:47,110 --> 00:11:49,320
for sines and cosines.
193
00:11:49,320 --> 00:11:50,960
From the sum formulas,
we worked out
194
00:11:50,960 --> 00:11:53,620
what the slope was everywhere.
195
00:11:53,620 --> 00:11:57,610
So we're following the same
procedure that we did before.
196
00:11:57,610 --> 00:12:00,960
But at this point we're stuck.
197
00:12:00,960 --> 00:12:04,660
We're stuck, because
that time using radians,
198
00:12:04,660 --> 00:12:07,140
this very clever idea
of radians in geometry,
199
00:12:07,140 --> 00:12:09,640
we were able to actually
figure out what the slope is.
200
00:12:09,640 --> 00:12:14,920
Whereas here, we're not so sure,
what M(2) is, for instance.
201
00:12:14,920 --> 00:12:17,200
We just don't know yet.
202
00:12:17,200 --> 00:12:22,790
So, the basic question that
we have to deal with right now
203
00:12:22,790 --> 00:12:32,060
is what is M(a)?
204
00:12:32,060 --> 00:12:34,680
That's what we're left with.
205
00:12:34,680 --> 00:12:42,990
And, the curious fact is
that the clever thing to do
206
00:12:42,990 --> 00:12:51,260
is to beg the question.
207
00:12:51,260 --> 00:12:54,730
So we're going to go through
a very circular route here.
208
00:12:54,730 --> 00:12:56,580
That is circuitous,
not circular.
209
00:12:56,580 --> 00:12:58,360
Circular is a bad word in math.
210
00:12:58,360 --> 00:13:00,380
That means that one
thing depends on another,
211
00:13:00,380 --> 00:13:03,220
and that depends on it,
and maybe both are wrong.
212
00:13:03,220 --> 00:13:07,460
Circuitous means, we're going
to be taking a roundabout route.
213
00:13:07,460 --> 00:13:10,384
And we're going to discover
that even though we refuse
214
00:13:10,384 --> 00:13:11,800
to answer this
question right now,
215
00:13:11,800 --> 00:13:14,841
we'll succeed in
answering it eventually.
216
00:13:14,841 --> 00:13:15,340
All right?
217
00:13:15,340 --> 00:13:18,340
So how are we going
to beg the question?
218
00:13:18,340 --> 00:13:20,150
What we're going
to say instead is
219
00:13:20,150 --> 00:13:30,650
we're going to define a
mystery base, or number e,
220
00:13:30,650 --> 00:13:45,790
as the unique number,
so that M(e) = 1.
221
00:13:45,790 --> 00:13:47,930
That's the trick that
we're going to use.
222
00:13:47,930 --> 00:13:50,610
We don't yet know what e
is, but we're just going
223
00:13:50,610 --> 00:13:53,900
to suppose that we have it.
224
00:13:53,900 --> 00:13:57,340
Now, I'm going to show you a
bunch of consequences of this,
225
00:13:57,340 --> 00:14:00,540
and also I have to persuade you
that it actually does exist.
226
00:14:00,540 --> 00:14:03,640
So first, let me explain what
the first consequence is.
227
00:14:03,640 --> 00:14:06,670
First of all, if M(e)
is 1, then if you
228
00:14:06,670 --> 00:14:09,540
look at this formula over here
and you write it down for e,
229
00:14:09,540 --> 00:14:13,650
you have something which
is a very usable formula.
230
00:14:13,650 --> 00:14:19,930
d/dx of e^x is just e^x.
231
00:14:19,930 --> 00:14:22,750
All right, so that's an
incredibly important formula
232
00:14:22,750 --> 00:14:24,210
which is the fundamental one.
233
00:14:24,210 --> 00:14:26,710
It's the only one you have to
remember from what we've done.
234
00:14:26,710 --> 00:14:28,251
So maybe I should
have highlighted it
235
00:14:28,251 --> 00:14:34,760
in several colors here.
236
00:14:34,760 --> 00:14:37,800
That's a big deal.
237
00:14:37,800 --> 00:14:40,630
Very happy.
238
00:14:40,630 --> 00:14:42,770
And again, let me
just emphasize,
239
00:14:42,770 --> 00:14:52,077
also that this is the one
which at x = 0 has slope 1.
240
00:14:52,077 --> 00:14:53,660
That's the way we
defined it, alright?
241
00:14:53,660 --> 00:15:00,640
So if you plug in x = 0 here on
the right hand side, you got 1.
242
00:15:00,640 --> 00:15:03,540
Slope 1 at x = 0.
243
00:15:03,540 --> 00:15:05,580
So that's great.
244
00:15:05,580 --> 00:15:07,980
Except of course, since
we don't know what e is,
245
00:15:07,980 --> 00:15:15,770
this is a little bit dicey.
246
00:15:15,770 --> 00:15:21,750
So, next even before
explaining what e is...
247
00:15:21,750 --> 00:15:23,440
In fact, we won't
get to what e really
248
00:15:23,440 --> 00:15:26,110
is until the very
end of this lecture.
249
00:15:26,110 --> 00:15:34,530
But I have to persuade
you why e exists.
250
00:15:34,530 --> 00:15:37,406
We have to have some
explanation for why
251
00:15:37,406 --> 00:15:40,740
we know there is such a number.
252
00:15:40,740 --> 00:15:44,100
Okay, so first of all, let
me start with the one that we
253
00:15:44,100 --> 00:15:46,970
supposedly know, which
is the function 2^x.
254
00:15:46,970 --> 00:15:49,710
We'll call it f(x) is 2^x.
255
00:15:49,710 --> 00:15:50,460
All right?
256
00:15:50,460 --> 00:15:51,820
So that's the first thing.
257
00:15:51,820 --> 00:15:54,460
And remember, that the
property that it had,
258
00:15:54,460 --> 00:15:58,170
was that f'(0) was M(2).
259
00:15:58,170 --> 00:16:04,430
That was the derivative of this
function, the slope at x = 0
260
00:16:04,430 --> 00:16:06,610
of the graph.
261
00:16:06,610 --> 00:16:09,870
Of the tangent line, that is.
262
00:16:09,870 --> 00:16:12,980
So now, what we're
going to consider
263
00:16:12,980 --> 00:16:16,880
is any kind of stretching.
264
00:16:16,880 --> 00:16:22,750
We're going to stretch this
function by a factor k.
265
00:16:22,750 --> 00:16:23,610
Any number k.
266
00:16:23,610 --> 00:16:29,090
So what we're going
to consider is f(kx).
267
00:16:29,090 --> 00:16:34,790
If you do that, that's
the same as 2^(kx).
268
00:16:34,790 --> 00:16:37,420
Right?
269
00:16:37,420 --> 00:16:41,030
But now if I use the second law
of exponents that I have over
270
00:16:41,030 --> 00:16:46,700
there, that's the same thing
as 2 to the k to the power x,
271
00:16:46,700 --> 00:16:51,110
which is the same
thing as some base b^x,
272
00:16:51,110 --> 00:16:55,420
where b is equal to-- Let's
write that down over here.
273
00:16:55,420 --> 00:16:55,940
b is 2^k.
274
00:16:59,081 --> 00:16:59,580
Right.
275
00:16:59,580 --> 00:17:03,630
So whatever it is, if I have
a different base which is
276
00:17:03,630 --> 00:17:07,740
expressed in terms of
2, of the form 2^k,
277
00:17:07,740 --> 00:17:14,110
then that new function is
described by this function
278
00:17:14,110 --> 00:17:17,700
f(kx), the stretch.
279
00:17:17,700 --> 00:17:20,730
So what happens when
you stretch a function?
280
00:17:20,730 --> 00:17:24,720
That's the same thing
as shrinking the x axis.
281
00:17:24,720 --> 00:17:30,035
So when k gets larger, this
corresponding point over here
282
00:17:30,035 --> 00:17:32,160
would be over here, and so
this corresponding point
283
00:17:32,160 --> 00:17:32,990
would be over here.
284
00:17:32,990 --> 00:17:38,910
So you shrink this picture,
and the slope here tilts up.
285
00:17:38,910 --> 00:17:43,140
So, as we increase k, the
slope gets steeper and steeper.
286
00:17:43,140 --> 00:17:47,570
Let's see that explicitly,
numerically, here.
287
00:17:47,570 --> 00:17:51,870
Explicitly, numerically, if
I take the derivative here...
288
00:17:51,870 --> 00:17:56,580
So the derivative with
respect to x of b^x,
289
00:17:56,580 --> 00:18:00,944
that's the chain rule, right?
290
00:18:00,944 --> 00:18:02,360
That's the derivative
with respect
291
00:18:02,360 --> 00:18:08,260
to x of f(kx), which is what?
292
00:18:08,260 --> 00:18:11,780
It's k times f'(kx).
293
00:18:11,780 --> 00:18:20,850
And so if we do it at 0,
we're just getting k times
294
00:18:20,850 --> 00:18:26,570
f'(0), which is k
times this M(2).
295
00:18:26,570 --> 00:18:31,390
So how is it exactly that
we cook up the right base b?
296
00:18:31,390 --> 00:18:40,260
So b = e when k =
1 over this number.
297
00:18:40,260 --> 00:18:44,279
In other words, we can pick all
possible slopes that we want.
298
00:18:44,279 --> 00:18:46,320
This just has the effect
of multiplying the slope
299
00:18:46,320 --> 00:18:47,550
by a factor.
300
00:18:47,550 --> 00:18:50,130
And we can shift the slope
at 0 however we want,
301
00:18:50,130 --> 00:18:56,240
and we're going to do it so
that the slope exactly matches
302
00:18:56,240 --> 00:18:58,150
1, the one that we want.
303
00:18:58,150 --> 00:18:59,580
We still don't know what k is.
304
00:18:59,580 --> 00:19:01,340
We still don't know what e is.
305
00:19:01,340 --> 00:19:04,940
But at least we know that
it's there somewhere.
306
00:19:04,940 --> 00:19:05,640
Yes?
307
00:19:05,640 --> 00:19:08,299
Student: How do you
know it's f(kx)?
308
00:19:08,299 --> 00:19:09,340
PROFESSOR: How do I know?
309
00:19:09,340 --> 00:19:13,440
Well, f(x) is 2^x.
310
00:19:13,440 --> 00:19:19,160
If f(x) is 2^x, then the
formula for f(kx) is this.
311
00:19:19,160 --> 00:19:23,060
I've decided what f(x)
is, so therefore there's
312
00:19:23,060 --> 00:19:25,120
a formula for f(kx).
313
00:19:25,120 --> 00:19:26,609
And furthermore,
by the chain rule,
314
00:19:26,609 --> 00:19:28,150
there's a formula
for the derivative.
315
00:19:28,150 --> 00:19:33,930
And it's k times
the derivative of f.
316
00:19:33,930 --> 00:19:35,150
So again, scaling does this.
317
00:19:35,150 --> 00:19:37,890
By the way, we did
exactly the same thing
318
00:19:37,890 --> 00:19:39,920
with the sine and
cosine function.
319
00:19:39,920 --> 00:19:41,650
If you think of
the sine function
320
00:19:41,650 --> 00:19:44,660
here, let me just
remind you here,
321
00:19:44,660 --> 00:19:46,310
what happens with
the chain rule,
322
00:19:46,310 --> 00:19:51,720
you get k times cosine k t here.
323
00:19:51,720 --> 00:19:55,340
So the fact that we set things
up beautifully with radians
324
00:19:55,340 --> 00:19:58,650
that this thing is, but we could
change the scale to anything,
325
00:19:58,650 --> 00:20:02,340
such as degrees, by the
appropriate factor k.
326
00:20:02,340 --> 00:20:05,320
And then there would be
this scale factor shift
327
00:20:05,320 --> 00:20:07,500
of the derivative formulas.
328
00:20:07,500 --> 00:20:09,500
Of course, the one with
radians is the easy one,
329
00:20:09,500 --> 00:20:11,130
because the factor is 1.
330
00:20:11,130 --> 00:20:13,850
The one with
degrees is horrible,
331
00:20:13,850 --> 00:20:20,085
because the factor is some
crazy number like 180 over pi,
332
00:20:20,085 --> 00:20:22,410
or something like that.
333
00:20:22,410 --> 00:20:25,880
Okay, so there's
something going on here
334
00:20:25,880 --> 00:20:30,420
which is exactly the same
as that kind of re-scaling.
335
00:20:30,420 --> 00:20:37,040
So, so far we've got only one
formula which is a keeper here.
336
00:20:37,040 --> 00:20:38,810
This one.
337
00:20:38,810 --> 00:20:40,870
We have a preliminary
formula that we still
338
00:20:40,870 --> 00:20:42,510
haven't completely
explained which
339
00:20:42,510 --> 00:20:45,820
has a little wavy line there.
340
00:20:45,820 --> 00:20:49,260
And we have to fit all
these things together.
341
00:20:49,260 --> 00:20:52,260
Okay, so now to
fit them together,
342
00:20:52,260 --> 00:21:11,450
I need to introduce
the natural log.
343
00:21:11,450 --> 00:21:21,590
So the natural log is
denoted this way, ln(x).
344
00:21:21,590 --> 00:21:24,570
So maybe I'll call
it a new letter name,
345
00:21:24,570 --> 00:21:28,470
we'll call it w = ln x here.
346
00:21:28,470 --> 00:21:32,040
But if we were reversing
things, if we started out with
347
00:21:32,040 --> 00:21:37,880
a function y = e^x , the
property that it would have is
348
00:21:37,880 --> 00:21:41,030
that it's the inverse
function of e^x.
349
00:21:41,030 --> 00:21:45,870
So it has the property that
the log of y is equal to x.
350
00:21:45,870 --> 00:21:46,370
Right?
351
00:21:46,370 --> 00:21:58,910
So this defines the log.
352
00:21:58,910 --> 00:22:01,790
Now the logarithm has
a bunch of properties
353
00:22:01,790 --> 00:22:04,270
and they come from the
exponential properties
354
00:22:04,270 --> 00:22:04,940
in principle.
355
00:22:04,940 --> 00:22:07,500
You remember these.
356
00:22:07,500 --> 00:22:10,379
And I'm just going to
remind you of them.
357
00:22:10,379 --> 00:22:12,420
So the main one that I
just want to remind you of
358
00:22:12,420 --> 00:22:21,100
is that the logarithm
of x_1 * x_2
359
00:22:21,100 --> 00:22:28,130
is equal to the logarithm of
x_1 plus the logarithm of x_2.
360
00:22:28,130 --> 00:22:32,170
And maybe a few more are
worth reminding you of.
361
00:22:32,170 --> 00:22:37,120
One is that the
logarithm of 1 is 0.
362
00:22:37,120 --> 00:22:43,310
A second is that the
logarithm of e is 1.
363
00:22:43,310 --> 00:22:43,840
All right?
364
00:22:43,840 --> 00:22:47,160
So these correspond to the
inverse relationships here.
365
00:22:47,160 --> 00:22:51,170
If I plug in here,
x = 0 and x = 1.
366
00:22:51,170 --> 00:22:56,650
If I plug in x = 0 and x = 1,
I get the corresponding numbers
367
00:22:56,650 --> 00:23:04,030
here: y = 1 and y = e.
368
00:23:04,030 --> 00:23:10,430
And maybe it would be worth
it to plot the picture once
369
00:23:10,430 --> 00:23:13,430
to reinforce this.
370
00:23:13,430 --> 00:23:16,620
So here I'll put them
on the same chart.
371
00:23:16,620 --> 00:23:20,200
If you have here e^x over here.
372
00:23:20,200 --> 00:23:21,790
It looks like this.
373
00:23:21,790 --> 00:23:28,610
Then the logarithm which I'll
maybe put in a different color.
374
00:23:28,610 --> 00:23:31,120
So this crosses at this
all-important point
375
00:23:31,120 --> 00:23:32,700
here, (0,1).
376
00:23:32,700 --> 00:23:35,260
And now in order to figure out
what the inverse function is,
377
00:23:35,260 --> 00:23:40,750
I have to take the flip
across the diagonal x = y.
378
00:23:40,750 --> 00:23:44,600
So that's this shape here,
going down like this.
379
00:23:44,600 --> 00:23:47,090
And here's the point (1, 0).
380
00:23:47,090 --> 00:23:50,700
So (1, 0) corresponds
to this identity here.
381
00:23:50,700 --> 00:23:53,000
But the log of 1 is 0.
382
00:23:53,000 --> 00:24:00,120
And notice, so this is
ln x, the graph of ln x.
383
00:24:00,120 --> 00:24:05,680
And notice it's only
defined for x positive,
384
00:24:05,680 --> 00:24:09,570
which corresponds to the fact
that e^x is always positive.
385
00:24:09,570 --> 00:24:15,130
So in other words, this white
curve is only above this axis,
386
00:24:15,130 --> 00:24:19,210
and the orange one
is to the right here.
387
00:24:19,210 --> 00:24:27,990
It's only defined
for x positive.
388
00:24:27,990 --> 00:24:31,740
Oh, one other thing I should
mention is the slope here is 1.
389
00:24:31,740 --> 00:24:35,380
And so the slope there
is also going to be 1.
390
00:24:35,380 --> 00:24:41,180
Now, what we're allowed to do
relatively easily, because we
391
00:24:41,180 --> 00:24:44,470
have the tools to do it, is
to compute the derivative
392
00:24:44,470 --> 00:24:49,960
of the logarithm.
393
00:24:49,960 --> 00:24:59,090
So to find the
derivative of a log,
394
00:24:59,090 --> 00:25:04,060
we're going to use
implicit differentiation.
395
00:25:04,060 --> 00:25:08,250
This is how we
find the derivative
396
00:25:08,250 --> 00:25:09,806
of any inverse function.
397
00:25:09,806 --> 00:25:11,180
So remember the
way that works is
398
00:25:11,180 --> 00:25:12,971
if you know the derivative
of the function,
399
00:25:12,971 --> 00:25:15,590
you can find the derivative
of the inverse function.
400
00:25:15,590 --> 00:25:18,280
And the mechanism
is the following:
401
00:25:18,280 --> 00:25:22,677
you write down here w = ln x.
402
00:25:22,677 --> 00:25:23,510
Here's the function.
403
00:25:23,510 --> 00:25:25,510
We're trying to find
the derivative of w.
404
00:25:25,510 --> 00:25:28,750
But now we don't know how to
differentiate this equation,
405
00:25:28,750 --> 00:25:38,200
but if we exponentiate it, so
that's the same thing as e^w =
406
00:25:38,200 --> 00:25:42,660
x.
407
00:25:42,660 --> 00:25:46,420
Because let's just
stick this in here.
408
00:25:46,420 --> 00:25:52,330
e^(ln x) = x.
409
00:25:52,330 --> 00:25:54,680
Now we can differentiate this.
410
00:25:54,680 --> 00:25:56,800
So let's do the
differentiation here.
411
00:25:56,800 --> 00:26:04,010
We have d/dx e^w is equal
to d/dx x, which is 1.
412
00:26:04,010 --> 00:26:06,030
And then this, by
the chain rule,
413
00:26:06,030 --> 00:26:11,450
is d/dw of e^w times dw/dx.
414
00:26:11,450 --> 00:26:14,560
The product of
these two factors.
415
00:26:14,560 --> 00:26:15,580
That's equal to 1.
416
00:26:15,580 --> 00:26:18,680
And now this guy,
the one little guy
417
00:26:18,680 --> 00:26:27,980
that we actually know and can
use, that's this guy here.
418
00:26:27,980 --> 00:26:33,800
So this is e^w times
dw/dx, which is 1.
419
00:26:33,800 --> 00:26:44,730
And so finally,
dw/dx = 1 / e^w .
420
00:26:44,730 --> 00:26:47,080
But what is that?
421
00:26:47,080 --> 00:26:48,250
It's x.
422
00:26:48,250 --> 00:26:50,740
So this is 1/x.
423
00:26:50,740 --> 00:26:53,140
So what we discovered
is, and now I
424
00:26:53,140 --> 00:26:57,000
get to put another
green guy around here,
425
00:26:57,000 --> 00:27:01,870
is that this is equal to 1/x.
426
00:27:01,870 --> 00:27:16,710
So alright, now we have two
companion formulas here.
427
00:27:16,710 --> 00:27:20,210
The rate of change
of ln x is 1/x.
428
00:27:20,210 --> 00:27:24,830
And the rate of change
of e^x is itself, is e^x.
429
00:27:24,830 --> 00:27:30,729
And it's time to
return to the problem
430
00:27:30,729 --> 00:27:32,770
that we were having a
little bit of trouble with,
431
00:27:32,770 --> 00:27:37,690
which is somewhat not explicit,
which is this M(a) times x.
432
00:27:37,690 --> 00:27:44,090
We want to now differentiate
a^x in general, not just e^x .
433
00:27:44,090 --> 00:27:47,140
So let's work that
out, and I want
434
00:27:47,140 --> 00:27:50,732
to explain it in
a couple of ways,
435
00:27:50,732 --> 00:27:52,440
so you're going to
have to remember this,
436
00:27:52,440 --> 00:27:55,530
because I'm going to erase it.
437
00:27:55,530 --> 00:28:01,780
But what I'd like you
to do is, so now I
438
00:28:01,780 --> 00:28:03,570
want to teach you
how to differentiate
439
00:28:03,570 --> 00:28:17,530
basically any exponential.
440
00:28:17,530 --> 00:28:31,580
So now to differentiate
any exponential.
441
00:28:31,580 --> 00:28:37,941
There are two methods.
442
00:28:37,941 --> 00:28:39,440
They're practically
the same method.
443
00:28:39,440 --> 00:28:41,480
They have the same
amount of arithmetic.
444
00:28:41,480 --> 00:28:45,610
You'll see both of them, and
they're equally valuable.
445
00:28:45,610 --> 00:28:48,150
So we're going to
just describe them.
446
00:28:48,150 --> 00:28:55,940
Method one I'm going to
illustrate on the function a^x.
447
00:28:55,940 --> 00:29:00,020
So we're interested
in differentiating
448
00:29:00,020 --> 00:29:04,280
this thing, exactly this problem
that I still didn't solve yet.
449
00:29:04,280 --> 00:29:05,080
Okay?
450
00:29:05,080 --> 00:29:06,950
So here it is.
451
00:29:06,950 --> 00:29:08,080
And here's the procedure.
452
00:29:08,080 --> 00:29:17,190
The procedure is to write, so
the method is to use base e,
453
00:29:17,190 --> 00:29:20,350
or convert to base e.
454
00:29:20,350 --> 00:29:22,430
So how do you convert to base e?
455
00:29:22,430 --> 00:29:27,660
Well, you write a^x
as e to some power.
456
00:29:27,660 --> 00:29:29,020
So what power is it?
457
00:29:29,020 --> 00:29:34,980
It's e to the power
ln a, to the power x.
458
00:29:34,980 --> 00:29:40,730
And that is just e^(x ln a).
459
00:29:40,730 --> 00:29:44,870
So we've made our
conversion now to base e.
460
00:29:44,870 --> 00:29:46,810
The exponential of something.
461
00:29:46,810 --> 00:29:50,410
So now I'm going to carry
out the differentiation.
462
00:29:50,410 --> 00:29:59,270
So d/dx of a^x is equal
to d/dx of e^(x ln a).
463
00:29:59,270 --> 00:30:05,970
And now, this is a step which
causes great confusion when
464
00:30:05,970 --> 00:30:06,870
you first see it.
465
00:30:06,870 --> 00:30:10,920
And you must get used to it,
because it's easy, not hard.
466
00:30:10,920 --> 00:30:13,450
Okay?
467
00:30:13,450 --> 00:30:18,660
The rate of change of
this with respect to x is,
468
00:30:18,660 --> 00:30:23,040
let me do it by analogy here.
469
00:30:23,040 --> 00:30:27,520
Because say I had e^(3x) and
I were differentiating it.
470
00:30:27,520 --> 00:30:31,600
The chain rule would
say that this is just 3,
471
00:30:31,600 --> 00:30:36,330
the rate of change of 3x with
respect to x times e^(3x).
472
00:30:36,330 --> 00:30:41,060
The rate of change of e to
the u with respect to u.
473
00:30:41,060 --> 00:30:43,500
So this is the
ordinary chain rule.
474
00:30:43,500 --> 00:30:47,610
And what we're doing up here
is exactly the same thing,
475
00:30:47,610 --> 00:30:50,270
because ln a, as
frightening as it
476
00:30:50,270 --> 00:30:54,690
looks, with all three letters
there, is just a fixed number.
477
00:30:54,690 --> 00:30:55,860
It's not moving.
478
00:30:55,860 --> 00:30:57,170
It's a constant.
479
00:30:57,170 --> 00:31:01,080
So the constant just
accelerates the rate of change
480
00:31:01,080 --> 00:31:04,980
by that factor, which is
what the chain rule is doing.
481
00:31:04,980 --> 00:31:11,830
So this is equal to
ln a times e^(x ln a).
482
00:31:11,830 --> 00:31:17,390
Same business here
with ln a replacing 3.
483
00:31:17,390 --> 00:31:19,790
So this is something you've
got to get used to in time
484
00:31:19,790 --> 00:31:21,789
for the exam, for instance,
because you're going
485
00:31:21,789 --> 00:31:25,360
to be doing a million of these.
486
00:31:25,360 --> 00:31:27,810
So do get used to it.
487
00:31:27,810 --> 00:31:29,230
So here's the formula.
488
00:31:29,230 --> 00:31:33,770
On the other hand, this
expression here was the same
489
00:31:33,770 --> 00:31:34,730
as a^x.
490
00:31:34,730 --> 00:31:39,200
So another way of writing this,
and I'll put this into a box,
491
00:31:39,200 --> 00:31:41,790
but actually I never
remember this particularly.
492
00:31:41,790 --> 00:31:48,650
I just re-derive it every time,
is that the derivative of a^x
493
00:31:48,650 --> 00:31:51,100
is equal to (ln a) a^x .
494
00:31:51,100 --> 00:31:56,930
Now I'm going to get rid
of what's underneath it.
495
00:31:56,930 --> 00:32:01,970
So this is another formula.
496
00:32:01,970 --> 00:32:05,500
So there's the formula I've
essentially finished here.
497
00:32:05,500 --> 00:32:11,190
And notice, what is
the magic number?
498
00:32:11,190 --> 00:32:16,089
The magic number is
the natural log of a.
499
00:32:16,089 --> 00:32:16,880
That's what it was.
500
00:32:16,880 --> 00:32:18,679
We didn't know what
it was in advance.
501
00:32:18,679 --> 00:32:19,470
This is what it is.
502
00:32:19,470 --> 00:32:21,450
It's the natural log of a.
503
00:32:21,450 --> 00:32:26,740
Let me emphasize to you
again, something about what's
504
00:32:26,740 --> 00:32:34,510
going on here, which has
to do with scale change.
505
00:32:34,510 --> 00:32:44,290
So, for example, the derivative
with respect to x of 2^x is (ln
506
00:32:44,290 --> 00:32:47,290
2) 2^x.
507
00:32:47,290 --> 00:32:49,080
The derivative
with respect to x,
508
00:32:49,080 --> 00:32:51,730
these are the two most obvious
bases that you might want
509
00:32:51,730 --> 00:32:56,760
to use, is ln 10 times 10^x .
510
00:32:56,760 --> 00:32:59,470
So one of the things that's
natural about the natural
511
00:32:59,470 --> 00:33:02,860
logarithm is that
even if we insisted
512
00:33:02,860 --> 00:33:07,410
that we must use base 2, or
that we must use base 10,
513
00:33:07,410 --> 00:33:11,360
we'd still be stuck
with natural logarithms.
514
00:33:11,360 --> 00:33:12,540
They come up naturally.
515
00:33:12,540 --> 00:33:14,780
They're the ones
which are independent
516
00:33:14,780 --> 00:33:20,079
of our human construct
of base 2 and base 10.
517
00:33:20,079 --> 00:33:21,620
The natural logarithm
is the one that
518
00:33:21,620 --> 00:33:25,360
comes up without reference.
519
00:33:25,360 --> 00:33:27,210
And we'll be mentioning
a few other ways
520
00:33:27,210 --> 00:33:31,110
in which it's natural later.
521
00:33:31,110 --> 00:33:34,750
So I told you about
this first method,
522
00:33:34,750 --> 00:33:41,730
now I want to tell you
about a second method here.
523
00:33:41,730 --> 00:34:05,700
So the second is called
logarithmic differentiation.
524
00:34:05,700 --> 00:34:07,770
So how does this work?
525
00:34:07,770 --> 00:34:10,930
Well, sometimes
you're having trouble
526
00:34:10,930 --> 00:34:17,640
differentiating a
function, and it's easier
527
00:34:17,640 --> 00:34:21,780
to differentiate its logarithm.
528
00:34:21,780 --> 00:34:23,830
That may seem peculiar,
but actually we'll
529
00:34:23,830 --> 00:34:26,640
give several examples where
this is clearly the case,
530
00:34:26,640 --> 00:34:28,560
that the logarithm is
easier to differentiate
531
00:34:28,560 --> 00:34:30,830
than the function.
532
00:34:30,830 --> 00:34:34,090
So it could be that this is an
easier quantity to understand.
533
00:34:34,090 --> 00:34:39,470
So we want to relate it
back to the function u.
534
00:34:39,470 --> 00:34:44,170
So I'm going to write it
a slightly different way.
535
00:34:44,170 --> 00:34:47,270
Let's write it in
terms of primes here.
536
00:34:47,270 --> 00:34:51,060
So the basic identity
is the chain rule again,
537
00:34:51,060 --> 00:34:52,730
and the derivative
of the logarithm,
538
00:34:52,730 --> 00:34:54,920
well maybe I'll write
it out this way first.
539
00:34:54,920 --> 00:35:01,720
So this would be d ln
u / du, times d/dx u.
540
00:35:05,120 --> 00:35:10,110
These are the two factors.
541
00:35:10,110 --> 00:35:12,040
And that's the same
thing, so remember
542
00:35:12,040 --> 00:35:14,140
what the derivative
of the logarithm is.
543
00:35:14,140 --> 00:35:17,820
This is 1/u.
544
00:35:17,820 --> 00:35:23,570
So here I have a 1/u,
and here I have a du/dx.
545
00:35:23,570 --> 00:35:28,850
So I'm going to encode this
on the next board here,
546
00:35:28,850 --> 00:35:31,290
which is sort of the main
formula you always need
547
00:35:31,290 --> 00:35:39,530
to remember, which is
that (ln u)' = u' / u.
548
00:35:39,530 --> 00:35:42,610
That's the one to remember here.
549
00:35:42,610 --> 00:35:47,320
STUDENT: [INAUDIBLE].
550
00:35:47,320 --> 00:35:51,870
PROFESSOR: The question is
how did I get this step here?
551
00:35:51,870 --> 00:35:58,500
So this is the chain rule.
552
00:35:58,500 --> 00:36:02,150
The rate of change of
ln u with respect to x
553
00:36:02,150 --> 00:36:04,610
is the rate of change
of ln u with respect u,
554
00:36:04,610 --> 00:36:07,730
times the rate of change
of u with respect to x.
555
00:36:07,730 --> 00:36:18,560
That's the chain rule.
556
00:36:18,560 --> 00:36:22,840
So now I've worked out
this identity here,
557
00:36:22,840 --> 00:36:30,730
and now let's show how it
handles this case, d/dx a^x.
558
00:36:30,730 --> 00:36:31,740
Let's do this one.
559
00:36:31,740 --> 00:36:39,570
So in order to get that
one, I would take u = a^x .
560
00:36:39,570 --> 00:36:51,460
And now let's just take a look
at what ln u is. ln u = x ln a.
561
00:36:51,460 --> 00:36:55,010
Now I claim that this is
pretty easy to differentiate.
562
00:36:55,010 --> 00:37:00,200
Again, it may seem hard, but
it's actually quite easy.
563
00:37:00,200 --> 00:37:04,590
So maybe somebody
can hazard a guess.
564
00:37:04,590 --> 00:37:11,530
What's the derivative of x ln a?
565
00:37:11,530 --> 00:37:14,870
It's just ln a.
566
00:37:14,870 --> 00:37:18,400
So this is the same thing that I
was talking about before, which
567
00:37:18,400 --> 00:37:21,420
is if you've got 3x,
and you're taking
568
00:37:21,420 --> 00:37:24,804
its derivative with respect
to x here, that's just 3.
569
00:37:24,804 --> 00:37:26,220
That's the kind
of thing you have.
570
00:37:26,220 --> 00:37:30,110
Again, don't be put off by this
massive piece of junk here.
571
00:37:30,110 --> 00:37:33,260
It's a constant.
572
00:37:33,260 --> 00:37:38,220
So again, keep that in mind.
573
00:37:38,220 --> 00:37:42,460
It comes up regularly in
this kind of question.
574
00:37:42,460 --> 00:37:46,980
So there's our formula, that the
logarithmic derivative is this.
575
00:37:46,980 --> 00:37:50,360
But let's just rewrite that.
576
00:37:50,360 --> 00:37:58,600
That's the same thing as u' / u,
which is (ln u)' = ln a, right?
577
00:37:58,600 --> 00:38:00,610
So this is our
differentiation formula.
578
00:38:00,610 --> 00:38:01,800
So here we have u'.
579
00:38:01,800 --> 00:38:07,460
u' is equal to u times ln a, if
I just multiply through by u.
580
00:38:07,460 --> 00:38:08,630
And that's what we wanted.
581
00:38:08,630 --> 00:38:16,660
That's d/dx a^x is equal to
ln a (I'll reverse the order
582
00:38:16,660 --> 00:38:24,640
of the two, which is
customary) times a^x.
583
00:38:24,640 --> 00:38:27,390
So this is the way that
logarithmic differentiation
584
00:38:27,390 --> 00:38:27,890
works.
585
00:38:27,890 --> 00:38:32,680
It's the same arithmetic
as the previous method,
586
00:38:32,680 --> 00:38:34,930
but we don't have to
convert to base e.
587
00:38:34,930 --> 00:38:37,829
We're just keeping
track of the exponents
588
00:38:37,829 --> 00:38:39,620
and doing differentiation
on the exponents,
589
00:38:39,620 --> 00:38:44,330
and multiplying
through at the end.
590
00:38:44,330 --> 00:38:49,660
Okay, so I'm going to do
two trickier examples, which
591
00:38:49,660 --> 00:39:02,400
illustrate logarithmic
differentiation.
592
00:39:02,400 --> 00:39:06,490
Again, these could be done
equally well by using base e,
593
00:39:06,490 --> 00:39:07,730
but I won't do it.
594
00:39:07,730 --> 00:39:12,120
Method one and method
two always both work.
595
00:39:12,120 --> 00:39:15,780
So here's a second
example: again this
596
00:39:15,780 --> 00:39:23,220
is a problem when you
have moving exponents.
597
00:39:23,220 --> 00:39:25,900
But this time, we're going
to complicate matters
598
00:39:25,900 --> 00:39:30,520
by having both a moving
exponent and a moving base.
599
00:39:30,520 --> 00:39:34,710
So we have a function u, which
is, well maybe I'll call it v,
600
00:39:34,710 --> 00:39:38,640
since we already had a
function u, which is x^x.
601
00:39:38,640 --> 00:39:41,670
A really complicated
looking function here.
602
00:39:41,670 --> 00:39:44,490
So again you can handle
this by converting
603
00:39:44,490 --> 00:39:47,220
to base e, method one.
604
00:39:47,220 --> 00:39:49,750
But we'll do the logarithmic
differentiation version,
605
00:39:49,750 --> 00:39:51,110
alright?
606
00:39:51,110 --> 00:39:59,310
So I take the logs
of both sides.
607
00:39:59,310 --> 00:40:04,370
And now I differentiate it.
608
00:40:04,370 --> 00:40:06,200
And now when I
differentiate this here,
609
00:40:06,200 --> 00:40:07,654
I have to use the product rule.
610
00:40:07,654 --> 00:40:09,570
This time, instead of
having ln a, a constant,
611
00:40:09,570 --> 00:40:10,980
I have a variable here.
612
00:40:10,980 --> 00:40:12,660
So I have two factors.
613
00:40:12,660 --> 00:40:15,416
I have ln x when I
differentiate with respect to x.
614
00:40:15,416 --> 00:40:19,410
When I differentiate with
respect to this factor here,
615
00:40:19,410 --> 00:40:21,800
I get that x times the
derivative of that,
616
00:40:21,800 --> 00:40:26,910
which is 1/x.
617
00:40:26,910 --> 00:40:29,160
So, here's my formula.
618
00:40:29,160 --> 00:40:30,430
Almost finished.
619
00:40:30,430 --> 00:40:34,701
So I have here v' / v. I'm going
to multiply these two things
620
00:40:34,701 --> 00:40:35,200
together.
621
00:40:35,200 --> 00:40:37,741
I'll put it on the other side,
because I don't want to get it
622
00:40:37,741 --> 00:40:45,340
mixed up with
ln(x+1), the quantity.
623
00:40:45,340 --> 00:40:47,100
And now I'm almost done.
624
00:40:47,100 --> 00:41:02,110
I have v' = v (1 + ln x), and
that's just d/dx x^x = x^x (1 +
625
00:41:02,110 --> 00:41:04,310
ln x).
626
00:41:04,310 --> 00:41:13,450
That's it.
627
00:41:13,450 --> 00:41:31,989
So these two methods always
work for moving exponents.
628
00:41:31,989 --> 00:41:33,530
So the next thing
that I'd like to do
629
00:41:33,530 --> 00:41:36,220
is another fairly
tricky example.
630
00:41:36,220 --> 00:41:45,880
And this one is not strictly
speaking within calculus.
631
00:41:45,880 --> 00:41:48,800
Although we're going to use the
tools that we just described
632
00:41:48,800 --> 00:41:52,820
to carry it out, in fact
it will use some calculus
633
00:41:52,820 --> 00:41:55,990
in the very end.
634
00:41:55,990 --> 00:41:59,530
And what I'm going to do is
I'm going to evaluate the limit
635
00:41:59,530 --> 00:42:02,020
as n goes to infinity
of (1 + 1/n)^n.
636
00:42:11,430 --> 00:42:16,170
So now, the reason why I want
to discuss this is, is it
637
00:42:16,170 --> 00:42:18,460
turns out to have a
very interesting answer.
638
00:42:18,460 --> 00:42:23,520
And it's a problem that
you can approach exactly
639
00:42:23,520 --> 00:42:24,490
by this method.
640
00:42:24,490 --> 00:42:28,580
And the reason is that
it has a moving exponent.
641
00:42:28,580 --> 00:42:30,840
The exponent n here is changing.
642
00:42:30,840 --> 00:42:33,570
And so if you want to keep track
of that, a good way to do that
643
00:42:33,570 --> 00:42:36,740
is to use logarithms.
644
00:42:36,740 --> 00:42:38,550
So in order to figure
out this limit,
645
00:42:38,550 --> 00:42:40,040
we're going to
take the log of it
646
00:42:40,040 --> 00:42:41,775
and figure out what
the limit of the log
647
00:42:41,775 --> 00:42:43,280
is, instead of the
log of the limit.
648
00:42:43,280 --> 00:42:44,990
Those will be the same thing.
649
00:42:44,990 --> 00:42:48,850
So we're going to take the
natural log of this quantity
650
00:42:48,850 --> 00:42:56,200
here, and that's n ln(1 + 1/n).
651
00:43:02,640 --> 00:43:06,370
And now I'm going
to rewrite this
652
00:43:06,370 --> 00:43:10,960
in a form which will make
it more recognizable,
653
00:43:10,960 --> 00:43:20,040
so what I'd like to do
is I'm going to write n,
654
00:43:20,040 --> 00:43:24,730
or maybe I should say it this
way: delta x is equal to 1/n.
655
00:43:24,730 --> 00:43:29,880
So if n is going to
infinity, then this delta x
656
00:43:29,880 --> 00:43:33,700
is going to be going to 0.
657
00:43:33,700 --> 00:43:37,500
So this is more familiar
territory for us in this class,
658
00:43:37,500 --> 00:43:38,560
anyway.
659
00:43:38,560 --> 00:43:40,370
So let's rewrite it.
660
00:43:40,370 --> 00:43:42,980
So here, we have 1 over delta x.
661
00:43:42,980 --> 00:43:46,680
And then that is multiplied
by ln(1 + delta x).
662
00:43:50,150 --> 00:43:54,860
So n is the
reciprocal of delta x.
663
00:43:54,860 --> 00:43:58,330
Now I want to change this
in a very, very minor way.
664
00:43:58,330 --> 00:44:01,150
I'm going to subtract 0 from it.
665
00:44:01,150 --> 00:44:02,470
So that's the same thing.
666
00:44:02,470 --> 00:44:06,420
So what I'm going to do is I'm
going to subtract ln 1 from it.
667
00:44:06,420 --> 00:44:08,260
That's just equal to 0.
668
00:44:08,260 --> 00:44:10,210
So this is not a
problem, and I'll
669
00:44:10,210 --> 00:44:14,800
put some parentheses around it.
670
00:44:14,800 --> 00:44:18,080
Now you're supposed to
recognize, all of a sudden,
671
00:44:18,080 --> 00:44:21,280
what pattern this fits into.
672
00:44:21,280 --> 00:44:25,570
This is the thing which we
need to calculate in order
673
00:44:25,570 --> 00:44:30,810
to calculate the derivative
of the log function.
674
00:44:30,810 --> 00:44:33,540
So this is, in
the limit as delta
675
00:44:33,540 --> 00:44:39,300
x goes to 0, equal to
the derivative of ln x.
676
00:44:39,300 --> 00:44:39,890
Where?
677
00:44:39,890 --> 00:44:43,940
Well the base point is x=1.
678
00:44:43,940 --> 00:44:45,710
That's where we're
evaluating it.
679
00:44:45,710 --> 00:44:46,730
That's the x_0.
680
00:44:46,730 --> 00:44:49,380
That's the base value.
681
00:44:49,380 --> 00:44:51,000
So this is the
difference quotient.
682
00:44:51,000 --> 00:44:52,430
That's exactly what it is.
683
00:44:52,430 --> 00:44:57,630
And so this by definition
tends to the limit here.
684
00:44:57,630 --> 00:45:01,470
But we know what the derivative
of the log function is.
685
00:45:01,470 --> 00:45:03,520
The derivative of the
log function is 1/x.
686
00:45:09,110 --> 00:45:17,470
So this limit is 1.
687
00:45:17,470 --> 00:45:18,500
So we got it.
688
00:45:18,500 --> 00:45:19,730
We got the limit.
689
00:45:19,730 --> 00:45:22,050
And now we just have
to work backwards
690
00:45:22,050 --> 00:45:34,010
to figure out what this limit
that we've got over here is.
691
00:45:34,010 --> 00:45:37,130
So let's do that.
692
00:45:37,130 --> 00:45:38,260
So let's see here.
693
00:45:38,260 --> 00:45:40,570
The log approached 1.
694
00:45:40,570 --> 00:45:45,960
So the limit as n goes to
infinity of (1 + 1/n)^n.
695
00:45:49,530 --> 00:45:51,880
So sorry, the log of this.
696
00:45:51,880 --> 00:45:54,320
Yeah, let's write it this way.
697
00:45:54,320 --> 00:45:57,530
It's the same thing, as
well, the thing that we know
698
00:45:57,530 --> 00:46:00,610
is the log of this.
699
00:46:00,610 --> 00:46:04,700
1 plus 1 over n to the n.
700
00:46:04,700 --> 00:46:06,390
And goes to infinity.
701
00:46:06,390 --> 00:46:08,190
That's the one that
we just figured out.
702
00:46:08,190 --> 00:46:11,380
But now this thing is
the exponential of that.
703
00:46:11,380 --> 00:46:16,550
So it's really e
to this power here.
704
00:46:16,550 --> 00:46:19,060
So this guy is the
same as the limit
705
00:46:19,060 --> 00:46:21,799
of the log of the limit of the
thing, which is the same as log
706
00:46:21,799 --> 00:46:22,340
of the limit.
707
00:46:22,340 --> 00:46:25,268
The limit of the log and the
log of the limit are the same.
708
00:46:25,268 --> 00:46:32,260
log lim equals lim log.
709
00:46:32,260 --> 00:46:33,754
Okay, so I take
the logarithm, then
710
00:46:33,754 --> 00:46:35,170
I'm going to take
the exponential.
711
00:46:35,170 --> 00:46:37,800
That just undoes
what I did before.
712
00:46:37,800 --> 00:46:41,910
And so this limit is
just 1, so this is e^1.
713
00:46:41,910 --> 00:46:52,000
And so the limit that we
want here is equal to e.
714
00:46:52,000 --> 00:46:56,790
So I claim that with this
step, we've actually closed
715
00:46:56,790 --> 00:46:58,270
the loop, finally.
716
00:46:58,270 --> 00:47:03,620
Because we have an honest
numerical way to calculate e.
717
00:47:03,620 --> 00:47:04,140
The first.
718
00:47:04,140 --> 00:47:05,400
There are many such.
719
00:47:05,400 --> 00:47:07,640
But this one is a perfectly
honest numerical way
720
00:47:07,640 --> 00:47:08,680
to calculate e.
721
00:47:08,680 --> 00:47:09,950
We had this thing.
722
00:47:09,950 --> 00:47:12,170
We didn't know
exactly what it was.
723
00:47:12,170 --> 00:47:14,872
It was this M(e), there was
M(a), the logarithm, and so on.
724
00:47:14,872 --> 00:47:15,830
We have all that stuff.
725
00:47:15,830 --> 00:47:18,920
But we really need to nail
down what this number e is.
726
00:47:18,920 --> 00:47:20,920
And this is telling
us, if you take
727
00:47:20,920 --> 00:47:25,620
for example 1 plus 1 over 100
to the 100th power, that's
728
00:47:25,620 --> 00:47:28,910
going to be a very good,
perfectly decent anyway,
729
00:47:28,910 --> 00:47:30,730
approximation to e.
730
00:47:30,730 --> 00:47:36,330
So this is a numerical
approximation,
731
00:47:36,330 --> 00:47:39,320
which is all we can
ever do with just
732
00:47:39,320 --> 00:47:42,700
this kind of irrational number.
733
00:47:42,700 --> 00:47:46,270
And so that closes
the loop, and we now
734
00:47:46,270 --> 00:47:49,452
have a coherent
family of functions,
735
00:47:49,452 --> 00:47:51,660
which are actually well
defined and for which we have
736
00:47:51,660 --> 00:47:54,550
practical methods to calculate.
737
00:47:54,550 --> 00:47:56,312
Okay, see you next time.