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PROFESSOR: All right, so
let's begin lecture six.
00:00:27.060 --> 00:00:44.630
We're talking today about
exponentials and logarithms.
00:00:44.630 --> 00:00:49.390
And these are the last functions
that I need to introduce,
00:00:49.390 --> 00:00:50.950
the last standard
functions that we
00:00:50.950 --> 00:00:55.110
need to connect with calculus,
that you've learned about.
00:00:55.110 --> 00:00:58.230
And they're certainly as
fundamental, if not more so,
00:00:58.230 --> 00:01:00.830
than trigonometric functions.
00:01:00.830 --> 00:01:04.960
So first of all, we'll
start out with a number,
00:01:04.960 --> 00:01:09.030
a, which is positive, which
is usually called a base.
00:01:09.030 --> 00:01:12.790
And then we have
these properties that
00:01:12.790 --> 00:01:14.870
a to the power 0 is always 1.
00:01:14.870 --> 00:01:17.180
That's how we get started.
00:01:17.180 --> 00:01:21.270
And a^1 is a.
00:01:21.270 --> 00:01:24.030
And of course a^2,
not surprisingly,
00:01:24.030 --> 00:01:26.200
is a times a, etc.
00:01:26.200 --> 00:01:35.400
And the general rule is that
a^(x_1 + x_2) is a^(x_1) times
00:01:35.400 --> 00:01:36.970
a^(x_2).
00:01:36.970 --> 00:01:41.440
So this is the basic rule of
exponents, and with these two
00:01:41.440 --> 00:01:49.070
initial properties, that defines
the exponential function.
00:01:49.070 --> 00:01:52.220
And then there's an
additional property,
00:01:52.220 --> 00:01:54.360
which is deduced
from these, which
00:01:54.360 --> 00:01:59.020
is the composition of
exponential functions, which
00:01:59.020 --> 00:02:03.330
is that you take a to the
x_1 power, to the x_2 power.
00:02:03.330 --> 00:02:08.390
Then that turns out to be
a to the x_1 times x_2.
00:02:08.390 --> 00:02:11.540
So that's an additional property
that we'll take for granted,
00:02:11.540 --> 00:02:14.140
which you learned
in high school.
00:02:14.140 --> 00:02:22.650
Now, in order to understand
what all the values of a^x are,
00:02:22.650 --> 00:02:28.620
we need to first remember that
if you're taking a rational
00:02:28.620 --> 00:02:34.350
power that it's the ratio
of two integers power of a.
00:02:34.350 --> 00:02:37.190
That's going to be a^m, and
then we're going to have to take
00:02:37.190 --> 00:02:39.440
the nth root of that.
00:02:39.440 --> 00:02:40.840
So that's the definition.
00:02:40.840 --> 00:02:45.050
And then, when
you're defining a^x,
00:02:45.050 --> 00:03:00.370
so a^x is defined for
all x by filling in.
00:03:00.370 --> 00:03:03.480
So I'm gonna use that
expression in quotation marks,
00:03:03.480 --> 00:03:09.930
"filling in" by continuity.
00:03:09.930 --> 00:03:11.540
This is really what
your calculator
00:03:11.540 --> 00:03:13.960
does when it gives
you a to the power x,
00:03:13.960 --> 00:03:17.540
because you can't even punch
in the square root of x.
00:03:17.540 --> 00:03:19.590
It doesn't really exist
on your calculator.
00:03:19.590 --> 00:03:21.230
There's some decimal expansion.
00:03:21.230 --> 00:03:24.400
So it takes the decimal
expansion to a certain length
00:03:24.400 --> 00:03:26.080
and spits out a
number which is pretty
00:03:26.080 --> 00:03:28.120
close to the correct answer.
00:03:28.120 --> 00:03:31.550
But indeed, in theory,
there is an a to the power
00:03:31.550 --> 00:03:33.710
square root of 2, even
though the square root of 2
00:03:33.710 --> 00:03:34.740
is irrational.
00:03:34.740 --> 00:03:37.540
And there's a to
the pi and so forth.
00:03:37.540 --> 00:03:41.000
All right, so that's the
exponential function,
00:03:41.000 --> 00:03:46.830
and let's draw a picture of one.
00:03:46.830 --> 00:03:52.230
So we'll try, say y = 2^x here.
00:03:52.230 --> 00:03:55.180
And I'm not going to draw
such a careful graph,
00:03:55.180 --> 00:03:58.630
but let's just plot the
most important point, which
00:03:58.630 --> 00:04:01.360
is the point (0,1).
00:04:01.360 --> 00:04:04.510
That's 2^0, which is 1.
00:04:04.510 --> 00:04:08.940
And then maybe we'll go
back up here to -1 here.
00:04:08.940 --> 00:04:13.860
And 2 to the -1 is
this point here.
00:04:13.860 --> 00:04:18.990
This is (-1, 1/2),
the reciprocal.
00:04:18.990 --> 00:04:23.460
And over here, we have 1, and so
that goes all the way up to 2.
00:04:23.460 --> 00:04:26.870
And then exponentials
are remarkably fast.
00:04:26.870 --> 00:04:30.680
So it's off the board what
happens next out at 2.
00:04:30.680 --> 00:04:34.360
It's already above
my range here,
00:04:34.360 --> 00:04:37.970
but the graph looks
something like this.
00:04:37.970 --> 00:04:38.610
All right.
00:04:38.610 --> 00:04:41.200
Now I've just visually,
at least, graphically
00:04:41.200 --> 00:04:43.240
filled in all the
rest of the points.
00:04:43.240 --> 00:04:46.730
You have to imagine all these
rational numbers, and so forth.
00:04:46.730 --> 00:04:51.670
So this point here
would have been (1, 2).
00:04:51.670 --> 00:04:53.330
And so forth.
00:04:53.330 --> 00:04:54.610
All right?
00:04:54.610 --> 00:05:00.992
So that's not too far along.
00:05:00.992 --> 00:05:01.950
So now what's our goal?
00:05:01.950 --> 00:05:04.020
Well, obviously we want
to do calculus here.
00:05:04.020 --> 00:05:08.050
So our goal, here, for now -
and it's gonna take a while.
00:05:08.050 --> 00:05:10.220
We have to think
about it pretty hard.
00:05:10.220 --> 00:05:22.250
We have to calculate
what this derivative is.
00:05:22.250 --> 00:05:26.020
All right, so we'll get started.
00:05:26.020 --> 00:05:28.520
And the way we get
started is simply
00:05:28.520 --> 00:05:31.190
by plugging in the
definition of the derivative.
00:05:31.190 --> 00:05:34.760
The derivative is
the limit as delta
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,
00:05:41.690 --> 00:05:45.080
divided by delta x.
00:05:45.080 --> 00:05:50.320
So that's what it is.
00:05:50.320 --> 00:05:56.140
And now, the only step that
we can really perform here
00:05:56.140 --> 00:05:58.790
to make this is into
something a little bit simpler
00:05:58.790 --> 00:06:03.200
is to use this very first
rule that we have here.
00:06:03.200 --> 00:06:06.930
That the exponential of
the sum is the product
00:06:06.930 --> 00:06:08.070
of the exponentials.
00:06:08.070 --> 00:06:10.340
So we have here, a^x .
00:06:10.340 --> 00:06:15.750
So what I want to use is just
the property that a^(x + delta
00:06:15.750 --> 00:06:22.460
x) = a^x a^(delta x).
00:06:22.460 --> 00:06:26.880
And if I do that, I see that I
can factor out a common factor
00:06:26.880 --> 00:06:29.760
in the numerator, which is a^x.
00:06:29.760 --> 00:06:35.120
So we'll write this as the
limit as delta x goes to 0,
00:06:35.120 --> 00:06:41.300
of a to the x times this ratio,
now a to the delta x, minus 1,
00:06:41.300 --> 00:06:49.280
divided by delta x.
00:06:49.280 --> 00:06:50.160
So far, so good?
00:06:50.160 --> 00:06:53.640
We're actually almost to
some serious progress here.
00:06:53.640 --> 00:06:58.460
So there's one other
important conceptual step
00:06:58.460 --> 00:07:00.100
which we need to understand.
00:07:00.100 --> 00:07:03.010
And this is a
relatively simple one.
00:07:03.010 --> 00:07:05.340
We actually did this
before, by the way.
00:07:05.340 --> 00:07:08.162
We did this with
sines and cosines.
00:07:08.162 --> 00:07:09.870
The next thing I want
to point out to you
00:07:09.870 --> 00:07:15.680
is that you're used to thinking
of x as being the variable.
00:07:15.680 --> 00:07:18.140
And indeed, already
we were discussing
00:07:18.140 --> 00:07:20.130
x as being the variable
and a as being fixed.
00:07:20.130 --> 00:07:22.220
But for the purposes
of this limit,
00:07:22.220 --> 00:07:26.290
there's a different variable
that's moving. x is fixed
00:07:26.290 --> 00:07:29.340
and delta x is the
thing that's moving.
00:07:29.340 --> 00:07:33.440
So that means that this factor
here, which is a common factor,
00:07:33.440 --> 00:07:34.799
is constant.
00:07:34.799 --> 00:07:36.590
And we can just factor
it out of the limit.
00:07:36.590 --> 00:07:39.450
It doesn't affect
the limit at all.
00:07:39.450 --> 00:07:41.130
A constant times a
limit is the same
00:07:41.130 --> 00:07:44.580
as whether we multiply before
or after we take the limit.
00:07:44.580 --> 00:07:46.830
So I'm just going
to factor that out.
00:07:46.830 --> 00:07:49.110
So that's my next step here.
00:07:49.110 --> 00:07:53.450
a^x, and then I have the
limit delta x goes to 0
00:07:53.450 --> 00:07:59.830
of a to the delta x minus
1, divided by delta x.
00:07:59.830 --> 00:08:02.180
All right?
00:08:02.180 --> 00:08:03.910
And so what I have
here, so this is
00:08:03.910 --> 00:08:05.220
by definition the derivative.
00:08:05.220 --> 00:08:12.730
So here is d/dx of a^x, and it's
equal to this expression here.
00:08:12.730 --> 00:08:17.340
Now, I want to stare
at this expression,
00:08:17.340 --> 00:08:22.880
and see what it's telling
us, because it's telling us
00:08:22.880 --> 00:08:27.680
as much as we can get
so far, without some--
00:08:27.680 --> 00:08:34.810
So first let's just
look at what this says.
00:08:34.810 --> 00:08:40.260
So what it's saying is that the
derivative of a^x is a^x times
00:08:40.260 --> 00:08:42.820
something that we
don't yet know.
00:08:42.820 --> 00:08:46.060
And I'm going to call this
something, this mystery number,
00:08:46.060 --> 00:08:47.130
M(a).
00:08:47.130 --> 00:08:52.990
So I'm gonna make the label,
M(a) is equal to the limit
00:08:52.990 --> 00:08:56.640
as delta x goes to 0
of a to the delta x
00:08:56.640 --> 00:08:59.800
minus 1 divided by delta x.
00:08:59.800 --> 00:09:00.300
All right?
00:09:00.300 --> 00:09:08.870
So this is a definition.
00:09:08.870 --> 00:09:14.540
So this mystery number M(a)
has a geometric interpretation,
00:09:14.540 --> 00:09:16.000
as well.
00:09:16.000 --> 00:09:17.554
So let me describe that.
00:09:17.554 --> 00:09:18.970
It has a geometric
interpretation,
00:09:18.970 --> 00:09:20.678
and it's a very, very
significant number.
00:09:20.678 --> 00:09:22.200
So let's work out what that is.
00:09:22.200 --> 00:09:25.690
So first of all, let's rewrite
the expression in the box,
00:09:25.690 --> 00:09:28.470
using the shorthand
for this number.
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
00:09:33.840 --> 00:09:37.800
factor, which is
M(a), times a^x.
00:09:37.800 --> 00:09:43.450
So the derivative of the
exponential is this mystery
00:09:43.450 --> 00:09:44.790
number times a^x.
00:09:44.790 --> 00:09:48.870
So we've almost solved
the problem of finding
00:09:48.870 --> 00:09:50.560
the derivative of a^x.
00:09:50.560 --> 00:09:53.270
We just have to figure
out this one number, M(a),
00:09:53.270 --> 00:09:55.720
and we get the rest.
00:09:55.720 --> 00:10:01.780
So let me point out two more
things about this number, M(a).
00:10:01.780 --> 00:10:09.350
So first of all,
if I plug in x = 0,
00:10:09.350 --> 00:10:14.250
that's going to be
d/dx of a^x , at x = 0.
00:10:14.250 --> 00:10:19.150
According to this formula,
that's M(a) times a^0,
00:10:19.150 --> 00:10:21.370
which of course M(a).
00:10:21.370 --> 00:10:23.540
So what is M(a)?
00:10:23.540 --> 00:10:26.410
M(a) is the derivative
of this function at 0.
00:10:26.410 --> 00:10:39.790
So M(a) is the slope of
a^x at x = 0, of the graph.
00:10:39.790 --> 00:10:41.330
The graph of a^x at 0.
00:10:41.330 --> 00:10:46.170
So again over here, if
you looked at the picture.
00:10:46.170 --> 00:10:48.260
I'll draw the one
tangent line in here,
00:10:48.260 --> 00:10:50.640
which is this one here.
00:10:50.640 --> 00:11:00.050
And this thing has slope,
what we're calling M(2).
00:11:00.050 --> 00:11:02.491
So, if I graph the
function y = 2^x,
00:11:02.491 --> 00:11:03.740
I'll get a certain slope here.
00:11:03.740 --> 00:11:05.315
If I graph it with
a different base,
00:11:05.315 --> 00:11:07.590
I might get another slope.
00:11:07.590 --> 00:11:12.820
And what we got so far is
the following phenomenon:
00:11:12.820 --> 00:11:16.310
if we know this one number, if
we know the slope at this one
00:11:16.310 --> 00:11:18.940
place, we will be able to figure
out the formula for the slope
00:11:18.940 --> 00:11:23.320
everywhere else.
00:11:23.320 --> 00:11:25.690
Now, that's actually
exactly the same thing
00:11:25.690 --> 00:11:28.040
that we did for
sines and cosines.
00:11:28.040 --> 00:11:33.120
We knew the slope of the
sine and the cosine function
00:11:33.120 --> 00:11:35.900
at x = 0.
00:11:35.900 --> 00:11:37.450
The sine function had slope 1.
00:11:37.450 --> 00:11:39.470
The cosine function had slope 0.
00:11:39.470 --> 00:11:41.560
And then from the
sum formulas, well
00:11:41.560 --> 00:11:44.004
that's exactly this
kind of thing here,
00:11:44.004 --> 00:11:44.920
from the sum formulas.
00:11:44.920 --> 00:11:47.110
This sum formula, in fact
is easier than the ones
00:11:47.110 --> 00:11:49.320
for sines and cosines.
00:11:49.320 --> 00:11:50.960
From the sum formulas,
we worked out
00:11:50.960 --> 00:11:53.620
what the slope was everywhere.
00:11:53.620 --> 00:11:57.610
So we're following the same
procedure that we did before.
00:11:57.610 --> 00:12:00.960
But at this point we're stuck.
00:12:00.960 --> 00:12:04.660
We're stuck, because
that time using radians,
00:12:04.660 --> 00:12:07.140
this very clever idea
of radians in geometry,
00:12:07.140 --> 00:12:09.640
we were able to actually
figure out what the slope is.
00:12:09.640 --> 00:12:14.920
Whereas here, we're not so sure,
what M(2) is, for instance.
00:12:14.920 --> 00:12:17.200
We just don't know yet.
00:12:17.200 --> 00:12:22.790
So, the basic question that
we have to deal with right now
00:12:22.790 --> 00:12:32.060
is what is M(a)?
00:12:32.060 --> 00:12:34.680
That's what we're left with.
00:12:34.680 --> 00:12:42.990
And, the curious fact is
that the clever thing to do
00:12:42.990 --> 00:12:51.260
is to beg the question.
00:12:51.260 --> 00:12:54.730
So we're going to go through
a very circular route here.
00:12:54.730 --> 00:12:56.580
That is circuitous,
not circular.
00:12:56.580 --> 00:12:58.360
Circular is a bad word in math.
00:12:58.360 --> 00:13:00.380
That means that one
thing depends on another,
00:13:00.380 --> 00:13:03.220
and that depends on it,
and maybe both are wrong.
00:13:03.220 --> 00:13:07.460
Circuitous means, we're going
to be taking a roundabout route.
00:13:07.460 --> 00:13:10.384
And we're going to discover
that even though we refuse
00:13:10.384 --> 00:13:11.800
to answer this
question right now,
00:13:11.800 --> 00:13:14.841
we'll succeed in
answering it eventually.
00:13:14.841 --> 00:13:15.340
All right?
00:13:15.340 --> 00:13:18.340
So how are we going
to beg the question?
00:13:18.340 --> 00:13:20.150
What we're going
to say instead is
00:13:20.150 --> 00:13:30.650
we're going to define a
mystery base, or number e,
00:13:30.650 --> 00:13:45.790
as the unique number,
so that M(e) = 1.
00:13:45.790 --> 00:13:47.930
That's the trick that
we're going to use.
00:13:47.930 --> 00:13:50.610
We don't yet know what e
is, but we're just going
00:13:50.610 --> 00:13:53.900
to suppose that we have it.
00:13:53.900 --> 00:13:57.340
Now, I'm going to show you a
bunch of consequences of this,
00:13:57.340 --> 00:14:00.540
and also I have to persuade you
that it actually does exist.
00:14:00.540 --> 00:14:03.640
So first, let me explain what
the first consequence is.
00:14:03.640 --> 00:14:06.670
First of all, if M(e)
is 1, then if you
00:14:06.670 --> 00:14:09.540
look at this formula over here
and you write it down for e,
00:14:09.540 --> 00:14:13.650
you have something which
is a very usable formula.
00:14:13.650 --> 00:14:19.930
d/dx of e^x is just e^x.
00:14:19.930 --> 00:14:22.750
All right, so that's an
incredibly important formula
00:14:22.750 --> 00:14:24.210
which is the fundamental one.
00:14:24.210 --> 00:14:26.710
It's the only one you have to
remember from what we've done.
00:14:26.710 --> 00:14:28.251
So maybe I should
have highlighted it
00:14:28.251 --> 00:14:34.760
in several colors here.
00:14:34.760 --> 00:14:37.800
That's a big deal.
00:14:37.800 --> 00:14:40.630
Very happy.
00:14:40.630 --> 00:14:42.770
And again, let me
just emphasize,
00:14:42.770 --> 00:14:52.077
also that this is the one
which at x = 0 has slope 1.
00:14:52.077 --> 00:14:53.660
That's the way we
defined it, alright?
00:14:53.660 --> 00:15:00.640
So if you plug in x = 0 here on
the right hand side, you got 1.
00:15:00.640 --> 00:15:03.540
Slope 1 at x = 0.
00:15:03.540 --> 00:15:05.580
So that's great.
00:15:05.580 --> 00:15:07.980
Except of course, since
we don't know what e is,
00:15:07.980 --> 00:15:15.770
this is a little bit dicey.
00:15:15.770 --> 00:15:21.750
So, next even before
explaining what e is...
00:15:21.750 --> 00:15:23.440
In fact, we won't
get to what e really
00:15:23.440 --> 00:15:26.110
is until the very
end of this lecture.
00:15:26.110 --> 00:15:34.530
But I have to persuade
you why e exists.
00:15:34.530 --> 00:15:37.406
We have to have some
explanation for why
00:15:37.406 --> 00:15:40.740
we know there is such a number.
00:15:40.740 --> 00:15:44.100
Okay, so first of all, let
me start with the one that we
00:15:44.100 --> 00:15:46.970
supposedly know, which
is the function 2^x.
00:15:46.970 --> 00:15:49.710
We'll call it f(x) is 2^x.
00:15:49.710 --> 00:15:50.460
All right?
00:15:50.460 --> 00:15:51.820
So that's the first thing.
00:15:51.820 --> 00:15:54.460
And remember, that the
property that it had,
00:15:54.460 --> 00:15:58.170
was that f'(0) was M(2).
00:15:58.170 --> 00:16:04.430
That was the derivative of this
function, the slope at x = 0
00:16:04.430 --> 00:16:06.610
of the graph.
00:16:06.610 --> 00:16:09.870
Of the tangent line, that is.
00:16:09.870 --> 00:16:12.980
So now, what we're
going to consider
00:16:12.980 --> 00:16:16.880
is any kind of stretching.
00:16:16.880 --> 00:16:22.750
We're going to stretch this
function by a factor k.
00:16:22.750 --> 00:16:23.610
Any number k.
00:16:23.610 --> 00:16:29.090
So what we're going
to consider is f(kx).
00:16:29.090 --> 00:16:34.790
If you do that, that's
the same as 2^(kx).
00:16:34.790 --> 00:16:37.420
Right?
00:16:37.420 --> 00:16:41.030
But now if I use the second law
of exponents that I have over
00:16:41.030 --> 00:16:46.700
there, that's the same thing
as 2 to the k to the power x,
00:16:46.700 --> 00:16:51.110
which is the same
thing as some base b^x,
00:16:51.110 --> 00:16:55.420
where b is equal to-- Let's
write that down over here.
00:16:55.420 --> 00:16:55.940
b is 2^k.
00:16:59.081 --> 00:16:59.580
Right.
00:16:59.580 --> 00:17:03.630
So whatever it is, if I have
a different base which is
00:17:03.630 --> 00:17:07.740
expressed in terms of
2, of the form 2^k,
00:17:07.740 --> 00:17:14.110
then that new function is
described by this function
00:17:14.110 --> 00:17:17.700
f(kx), the stretch.
00:17:17.700 --> 00:17:20.730
So what happens when
you stretch a function?
00:17:20.730 --> 00:17:24.720
That's the same thing
as shrinking the x axis.
00:17:24.720 --> 00:17:30.035
So when k gets larger, this
corresponding point over here
00:17:30.035 --> 00:17:32.160
would be over here, and so
this corresponding point
00:17:32.160 --> 00:17:32.990
would be over here.
00:17:32.990 --> 00:17:38.910
So you shrink this picture,
and the slope here tilts up.
00:17:38.910 --> 00:17:43.140
So, as we increase k, the
slope gets steeper and steeper.
00:17:43.140 --> 00:17:47.570
Let's see that explicitly,
numerically, here.
00:17:47.570 --> 00:17:51.870
Explicitly, numerically, if
I take the derivative here...
00:17:51.870 --> 00:17:56.580
So the derivative with
respect to x of b^x,
00:17:56.580 --> 00:18:00.944
that's the chain rule, right?
00:18:00.944 --> 00:18:02.360
That's the derivative
with respect
00:18:02.360 --> 00:18:08.260
to x of f(kx), which is what?
00:18:08.260 --> 00:18:11.780
It's k times f'(kx).
00:18:11.780 --> 00:18:20.850
And so if we do it at 0,
we're just getting k times
00:18:20.850 --> 00:18:26.570
f'(0), which is k
times this M(2).
00:18:26.570 --> 00:18:31.390
So how is it exactly that
we cook up the right base b?
00:18:31.390 --> 00:18:40.260
So b = e when k =
1 over this number.
00:18:40.260 --> 00:18:44.279
In other words, we can pick all
possible slopes that we want.
00:18:44.279 --> 00:18:46.320
This just has the effect
of multiplying the slope
00:18:46.320 --> 00:18:47.550
by a factor.
00:18:47.550 --> 00:18:50.130
And we can shift the slope
at 0 however we want,
00:18:50.130 --> 00:18:56.240
and we're going to do it so
that the slope exactly matches
00:18:56.240 --> 00:18:58.150
1, the one that we want.
00:18:58.150 --> 00:18:59.580
We still don't know what k is.
00:18:59.580 --> 00:19:01.340
We still don't know what e is.
00:19:01.340 --> 00:19:04.940
But at least we know that
it's there somewhere.
00:19:04.940 --> 00:19:05.640
Yes?
00:19:05.640 --> 00:19:08.299
Student: How do you
know it's f(kx)?
00:19:08.299 --> 00:19:09.340
PROFESSOR: How do I know?
00:19:09.340 --> 00:19:13.440
Well, f(x) is 2^x.
00:19:13.440 --> 00:19:19.160
If f(x) is 2^x, then the
formula for f(kx) is this.
00:19:19.160 --> 00:19:23.060
I've decided what f(x)
is, so therefore there's
00:19:23.060 --> 00:19:25.120
a formula for f(kx).
00:19:25.120 --> 00:19:26.609
And furthermore,
by the chain rule,
00:19:26.609 --> 00:19:28.150
there's a formula
for the derivative.
00:19:28.150 --> 00:19:33.930
And it's k times
the derivative of f.
00:19:33.930 --> 00:19:35.150
So again, scaling does this.
00:19:35.150 --> 00:19:37.890
By the way, we did
exactly the same thing
00:19:37.890 --> 00:19:39.920
with the sine and
cosine function.
00:19:39.920 --> 00:19:41.650
If you think of
the sine function
00:19:41.650 --> 00:19:44.660
here, let me just
remind you here,
00:19:44.660 --> 00:19:46.310
what happens with
the chain rule,
00:19:46.310 --> 00:19:51.720
you get k times cosine k t here.
00:19:51.720 --> 00:19:55.340
So the fact that we set things
up beautifully with radians
00:19:55.340 --> 00:19:58.650
that this thing is, but we could
change the scale to anything,
00:19:58.650 --> 00:20:02.340
such as degrees, by the
appropriate factor k.
00:20:02.340 --> 00:20:05.320
And then there would be
this scale factor shift
00:20:05.320 --> 00:20:07.500
of the derivative formulas.
00:20:07.500 --> 00:20:09.500
Of course, the one with
radians is the easy one,
00:20:09.500 --> 00:20:11.130
because the factor is 1.
00:20:11.130 --> 00:20:13.850
The one with
degrees is horrible,
00:20:13.850 --> 00:20:20.085
because the factor is some
crazy number like 180 over pi,
00:20:20.085 --> 00:20:22.410
or something like that.
00:20:22.410 --> 00:20:25.880
Okay, so there's
something going on here
00:20:25.880 --> 00:20:30.420
which is exactly the same
as that kind of re-scaling.
00:20:30.420 --> 00:20:37.040
So, so far we've got only one
formula which is a keeper here.
00:20:37.040 --> 00:20:38.810
This one.
00:20:38.810 --> 00:20:40.870
We have a preliminary
formula that we still
00:20:40.870 --> 00:20:42.510
haven't completely
explained which
00:20:42.510 --> 00:20:45.820
has a little wavy line there.
00:20:45.820 --> 00:20:49.260
And we have to fit all
these things together.
00:20:49.260 --> 00:20:52.260
Okay, so now to
fit them together,
00:20:52.260 --> 00:21:11.450
I need to introduce
the natural log.
00:21:11.450 --> 00:21:21.590
So the natural log is
denoted this way, ln(x).
00:21:21.590 --> 00:21:24.570
So maybe I'll call
it a new letter name,
00:21:24.570 --> 00:21:28.470
we'll call it w = ln x here.
00:21:28.470 --> 00:21:32.040
But if we were reversing
things, if we started out with
00:21:32.040 --> 00:21:37.880
a function y = e^x , the
property that it would have is
00:21:37.880 --> 00:21:41.030
that it's the inverse
function of e^x.
00:21:41.030 --> 00:21:45.870
So it has the property that
the log of y is equal to x.
00:21:45.870 --> 00:21:46.370
Right?
00:21:46.370 --> 00:21:58.910
So this defines the log.
00:21:58.910 --> 00:22:01.790
Now the logarithm has
a bunch of properties
00:22:01.790 --> 00:22:04.270
and they come from the
exponential properties
00:22:04.270 --> 00:22:04.940
in principle.
00:22:04.940 --> 00:22:07.500
You remember these.
00:22:07.500 --> 00:22:10.379
And I'm just going to
remind you of them.
00:22:10.379 --> 00:22:12.420
So the main one that I
just want to remind you of
00:22:12.420 --> 00:22:21.100
is that the logarithm
of x_1 * x_2
00:22:21.100 --> 00:22:28.130
is equal to the logarithm of
x_1 plus the logarithm of x_2.
00:22:28.130 --> 00:22:32.170
And maybe a few more are
worth reminding you of.
00:22:32.170 --> 00:22:37.120
One is that the
logarithm of 1 is 0.
00:22:37.120 --> 00:22:43.310
A second is that the
logarithm of e is 1.
00:22:43.310 --> 00:22:43.840
All right?
00:22:43.840 --> 00:22:47.160
So these correspond to the
inverse relationships here.
00:22:47.160 --> 00:22:51.170
If I plug in here,
x = 0 and x = 1.
00:22:51.170 --> 00:22:56.650
If I plug in x = 0 and x = 1,
I get the corresponding numbers
00:22:56.650 --> 00:23:04.030
here: y = 1 and y = e.
00:23:04.030 --> 00:23:10.430
And maybe it would be worth
it to plot the picture once
00:23:10.430 --> 00:23:13.430
to reinforce this.
00:23:13.430 --> 00:23:16.620
So here I'll put them
on the same chart.
00:23:16.620 --> 00:23:20.200
If you have here e^x over here.
00:23:20.200 --> 00:23:21.790
It looks like this.
00:23:21.790 --> 00:23:28.610
Then the logarithm which I'll
maybe put in a different color.
00:23:28.610 --> 00:23:31.120
So this crosses at this
all-important point
00:23:31.120 --> 00:23:32.700
here, (0,1).
00:23:32.700 --> 00:23:35.260
And now in order to figure out
what the inverse function is,
00:23:35.260 --> 00:23:40.750
I have to take the flip
across the diagonal x = y.
00:23:40.750 --> 00:23:44.600
So that's this shape here,
going down like this.
00:23:44.600 --> 00:23:47.090
And here's the point (1, 0).
00:23:47.090 --> 00:23:50.700
So (1, 0) corresponds
to this identity here.
00:23:50.700 --> 00:23:53.000
But the log of 1 is 0.
00:23:53.000 --> 00:24:00.120
And notice, so this is
ln x, the graph of ln x.
00:24:00.120 --> 00:24:05.680
And notice it's only
defined for x positive,
00:24:05.680 --> 00:24:09.570
which corresponds to the fact
that e^x is always positive.
00:24:09.570 --> 00:24:15.130
So in other words, this white
curve is only above this axis,
00:24:15.130 --> 00:24:19.210
and the orange one
is to the right here.
00:24:19.210 --> 00:24:27.990
It's only defined
for x positive.
00:24:27.990 --> 00:24:31.740
Oh, one other thing I should
mention is the slope here is 1.
00:24:31.740 --> 00:24:35.380
And so the slope there
is also going to be 1.
00:24:35.380 --> 00:24:41.180
Now, what we're allowed to do
relatively easily, because we
00:24:41.180 --> 00:24:44.470
have the tools to do it, is
to compute the derivative
00:24:44.470 --> 00:24:49.960
of the logarithm.
00:24:49.960 --> 00:24:59.090
So to find the
derivative of a log,
00:24:59.090 --> 00:25:04.060
we're going to use
implicit differentiation.
00:25:04.060 --> 00:25:08.250
This is how we
find the derivative
00:25:08.250 --> 00:25:09.806
of any inverse function.
00:25:09.806 --> 00:25:11.180
So remember the
way that works is
00:25:11.180 --> 00:25:12.971
if you know the derivative
of the function,
00:25:12.971 --> 00:25:15.590
you can find the derivative
of the inverse function.
00:25:15.590 --> 00:25:18.280
And the mechanism
is the following:
00:25:18.280 --> 00:25:22.677
you write down here w = ln x.
00:25:22.677 --> 00:25:23.510
Here's the function.
00:25:23.510 --> 00:25:25.510
We're trying to find
the derivative of w.
00:25:25.510 --> 00:25:28.750
But now we don't know how to
differentiate this equation,
00:25:28.750 --> 00:25:38.200
but if we exponentiate it, so
that's the same thing as e^w =
00:25:38.200 --> 00:25:42.660
x.
00:25:42.660 --> 00:25:46.420
Because let's just
stick this in here.
00:25:46.420 --> 00:25:52.330
e^(ln x) = x.
00:25:52.330 --> 00:25:54.680
Now we can differentiate this.
00:25:54.680 --> 00:25:56.800
So let's do the
differentiation here.
00:25:56.800 --> 00:26:04.010
We have d/dx e^w is equal
to d/dx x, which is 1.
00:26:04.010 --> 00:26:06.030
And then this, by
the chain rule,
00:26:06.030 --> 00:26:11.450
is d/dw of e^w times dw/dx.
00:26:11.450 --> 00:26:14.560
The product of
these two factors.
00:26:14.560 --> 00:26:15.580
That's equal to 1.
00:26:15.580 --> 00:26:18.680
And now this guy,
the one little guy
00:26:18.680 --> 00:26:27.980
that we actually know and can
use, that's this guy here.
00:26:27.980 --> 00:26:33.800
So this is e^w times
dw/dx, which is 1.
00:26:33.800 --> 00:26:44.730
And so finally,
dw/dx = 1 / e^w .
00:26:44.730 --> 00:26:47.080
But what is that?
00:26:47.080 --> 00:26:48.250
It's x.
00:26:48.250 --> 00:26:50.740
So this is 1/x.
00:26:50.740 --> 00:26:53.140
So what we discovered
is, and now I
00:26:53.140 --> 00:26:57.000
get to put another
green guy around here,
00:26:57.000 --> 00:27:01.870
is that this is equal to 1/x.
00:27:01.870 --> 00:27:16.710
So alright, now we have two
companion formulas here.
00:27:16.710 --> 00:27:20.210
The rate of change
of ln x is 1/x.
00:27:20.210 --> 00:27:24.830
And the rate of change
of e^x is itself, is e^x.
00:27:24.830 --> 00:27:30.729
And it's time to
return to the problem
00:27:30.729 --> 00:27:32.770
that we were having a
little bit of trouble with,
00:27:32.770 --> 00:27:37.690
which is somewhat not explicit,
which is this M(a) times x.
00:27:37.690 --> 00:27:44.090
We want to now differentiate
a^x in general, not just e^x .
00:27:44.090 --> 00:27:47.140
So let's work that
out, and I want
00:27:47.140 --> 00:27:50.732
to explain it in
a couple of ways,
00:27:50.732 --> 00:27:52.440
so you're going to
have to remember this,
00:27:52.440 --> 00:27:55.530
because I'm going to erase it.
00:27:55.530 --> 00:28:01.780
But what I'd like you
to do is, so now I
00:28:01.780 --> 00:28:03.570
want to teach you
how to differentiate
00:28:03.570 --> 00:28:17.530
basically any exponential.
00:28:17.530 --> 00:28:31.580
So now to differentiate
any exponential.
00:28:31.580 --> 00:28:37.941
There are two methods.
00:28:37.941 --> 00:28:39.440
They're practically
the same method.
00:28:39.440 --> 00:28:41.480
They have the same
amount of arithmetic.
00:28:41.480 --> 00:28:45.610
You'll see both of them, and
they're equally valuable.
00:28:45.610 --> 00:28:48.150
So we're going to
just describe them.
00:28:48.150 --> 00:28:55.940
Method one I'm going to
illustrate on the function a^x.
00:28:55.940 --> 00:29:00.020
So we're interested
in differentiating
00:29:00.020 --> 00:29:04.280
this thing, exactly this problem
that I still didn't solve yet.
00:29:04.280 --> 00:29:05.080
Okay?
00:29:05.080 --> 00:29:06.950
So here it is.
00:29:06.950 --> 00:29:08.080
And here's the procedure.
00:29:08.080 --> 00:29:17.190
The procedure is to write, so
the method is to use base e,
00:29:17.190 --> 00:29:20.350
or convert to base e.
00:29:20.350 --> 00:29:22.430
So how do you convert to base e?
00:29:22.430 --> 00:29:27.660
Well, you write a^x
as e to some power.
00:29:27.660 --> 00:29:29.020
So what power is it?
00:29:29.020 --> 00:29:34.980
It's e to the power
ln a, to the power x.
00:29:34.980 --> 00:29:40.730
And that is just e^(x ln a).
00:29:40.730 --> 00:29:44.870
So we've made our
conversion now to base e.
00:29:44.870 --> 00:29:46.810
The exponential of something.
00:29:46.810 --> 00:29:50.410
So now I'm going to carry
out the differentiation.
00:29:50.410 --> 00:29:59.270
So d/dx of a^x is equal
to d/dx of e^(x ln a).
00:29:59.270 --> 00:30:05.970
And now, this is a step which
causes great confusion when
00:30:05.970 --> 00:30:06.870
you first see it.
00:30:06.870 --> 00:30:10.920
And you must get used to it,
because it's easy, not hard.
00:30:10.920 --> 00:30:13.450
Okay?
00:30:13.450 --> 00:30:18.660
The rate of change of
this with respect to x is,
00:30:18.660 --> 00:30:23.040
let me do it by analogy here.
00:30:23.040 --> 00:30:27.520
Because say I had e^(3x) and
I were differentiating it.
00:30:27.520 --> 00:30:31.600
The chain rule would
say that this is just 3,
00:30:31.600 --> 00:30:36.330
the rate of change of 3x with
respect to x times e^(3x).
00:30:36.330 --> 00:30:41.060
The rate of change of e to
the u with respect to u.
00:30:41.060 --> 00:30:43.500
So this is the
ordinary chain rule.
00:30:43.500 --> 00:30:47.610
And what we're doing up here
is exactly the same thing,
00:30:47.610 --> 00:30:50.270
because ln a, as
frightening as it
00:30:50.270 --> 00:30:54.690
looks, with all three letters
there, is just a fixed number.
00:30:54.690 --> 00:30:55.860
It's not moving.
00:30:55.860 --> 00:30:57.170
It's a constant.
00:30:57.170 --> 00:31:01.080
So the constant just
accelerates the rate of change
00:31:01.080 --> 00:31:04.980
by that factor, which is
what the chain rule is doing.
00:31:04.980 --> 00:31:11.830
So this is equal to
ln a times e^(x ln a).
00:31:11.830 --> 00:31:17.390
Same business here
with ln a replacing 3.
00:31:17.390 --> 00:31:19.790
So this is something you've
got to get used to in time
00:31:19.790 --> 00:31:21.789
for the exam, for instance,
because you're going
00:31:21.789 --> 00:31:25.360
to be doing a million of these.
00:31:25.360 --> 00:31:27.810
So do get used to it.
00:31:27.810 --> 00:31:29.230
So here's the formula.
00:31:29.230 --> 00:31:33.770
On the other hand, this
expression here was the same
00:31:33.770 --> 00:31:34.730
as a^x.
00:31:34.730 --> 00:31:39.200
So another way of writing this,
and I'll put this into a box,
00:31:39.200 --> 00:31:41.790
but actually I never
remember this particularly.
00:31:41.790 --> 00:31:48.650
I just re-derive it every time,
is that the derivative of a^x
00:31:48.650 --> 00:31:51.100
is equal to (ln a) a^x .
00:31:51.100 --> 00:31:56.930
Now I'm going to get rid
of what's underneath it.
00:31:56.930 --> 00:32:01.970
So this is another formula.
00:32:01.970 --> 00:32:05.500
So there's the formula I've
essentially finished here.
00:32:05.500 --> 00:32:11.190
And notice, what is
the magic number?
00:32:11.190 --> 00:32:16.089
The magic number is
the natural log of a.
00:32:16.089 --> 00:32:16.880
That's what it was.
00:32:16.880 --> 00:32:18.679
We didn't know what
it was in advance.
00:32:18.679 --> 00:32:19.470
This is what it is.
00:32:19.470 --> 00:32:21.450
It's the natural log of a.
00:32:21.450 --> 00:32:26.740
Let me emphasize to you
again, something about what's
00:32:26.740 --> 00:32:34.510
going on here, which has
to do with scale change.
00:32:34.510 --> 00:32:44.290
So, for example, the derivative
with respect to x of 2^x is (ln
00:32:44.290 --> 00:32:47.290
2) 2^x.
00:32:47.290 --> 00:32:49.080
The derivative
with respect to x,
00:32:49.080 --> 00:32:51.730
these are the two most obvious
bases that you might want
00:32:51.730 --> 00:32:56.760
to use, is ln 10 times 10^x .
00:32:56.760 --> 00:32:59.470
So one of the things that's
natural about the natural
00:32:59.470 --> 00:33:02.860
logarithm is that
even if we insisted
00:33:02.860 --> 00:33:07.410
that we must use base 2, or
that we must use base 10,
00:33:07.410 --> 00:33:11.360
we'd still be stuck
with natural logarithms.
00:33:11.360 --> 00:33:12.540
They come up naturally.
00:33:12.540 --> 00:33:14.780
They're the ones
which are independent
00:33:14.780 --> 00:33:20.079
of our human construct
of base 2 and base 10.
00:33:20.079 --> 00:33:21.620
The natural logarithm
is the one that
00:33:21.620 --> 00:33:25.360
comes up without reference.
00:33:25.360 --> 00:33:27.210
And we'll be mentioning
a few other ways
00:33:27.210 --> 00:33:31.110
in which it's natural later.
00:33:31.110 --> 00:33:34.750
So I told you about
this first method,
00:33:34.750 --> 00:33:41.730
now I want to tell you
about a second method here.
00:33:41.730 --> 00:34:05.700
So the second is called
logarithmic differentiation.
00:34:05.700 --> 00:34:07.770
So how does this work?
00:34:07.770 --> 00:34:10.930
Well, sometimes
you're having trouble
00:34:10.930 --> 00:34:17.640
differentiating a
function, and it's easier
00:34:17.640 --> 00:34:21.780
to differentiate its logarithm.
00:34:21.780 --> 00:34:23.830
That may seem peculiar,
but actually we'll
00:34:23.830 --> 00:34:26.640
give several examples where
this is clearly the case,
00:34:26.640 --> 00:34:28.560
that the logarithm is
easier to differentiate
00:34:28.560 --> 00:34:30.830
than the function.
00:34:30.830 --> 00:34:34.090
So it could be that this is an
easier quantity to understand.
00:34:34.090 --> 00:34:39.470
So we want to relate it
back to the function u.
00:34:39.470 --> 00:34:44.170
So I'm going to write it
a slightly different way.
00:34:44.170 --> 00:34:47.270
Let's write it in
terms of primes here.
00:34:47.270 --> 00:34:51.060
So the basic identity
is the chain rule again,
00:34:51.060 --> 00:34:52.730
and the derivative
of the logarithm,
00:34:52.730 --> 00:34:54.920
well maybe I'll write
it out this way first.
00:34:54.920 --> 00:35:01.720
So this would be d ln
u / du, times d/dx u.
00:35:05.120 --> 00:35:10.110
These are the two factors.
00:35:10.110 --> 00:35:12.040
And that's the same
thing, so remember
00:35:12.040 --> 00:35:14.140
what the derivative
of the logarithm is.
00:35:14.140 --> 00:35:17.820
This is 1/u.
00:35:17.820 --> 00:35:23.570
So here I have a 1/u,
and here I have a du/dx.
00:35:23.570 --> 00:35:28.850
So I'm going to encode this
on the next board here,
00:35:28.850 --> 00:35:31.290
which is sort of the main
formula you always need
00:35:31.290 --> 00:35:39.530
to remember, which is
that (ln u)' = u' / u.
00:35:39.530 --> 00:35:42.610
That's the one to remember here.
00:35:42.610 --> 00:35:47.320
STUDENT: [INAUDIBLE].
00:35:47.320 --> 00:35:51.870
PROFESSOR: The question is
how did I get this step here?
00:35:51.870 --> 00:35:58.500
So this is the chain rule.
00:35:58.500 --> 00:36:02.150
The rate of change of
ln u with respect to x
00:36:02.150 --> 00:36:04.610
is the rate of change
of ln u with respect u,
00:36:04.610 --> 00:36:07.730
times the rate of change
of u with respect to x.
00:36:07.730 --> 00:36:18.560
That's the chain rule.
00:36:18.560 --> 00:36:22.840
So now I've worked out
this identity here,
00:36:22.840 --> 00:36:30.730
and now let's show how it
handles this case, d/dx a^x.
00:36:30.730 --> 00:36:31.740
Let's do this one.
00:36:31.740 --> 00:36:39.570
So in order to get that
one, I would take u = a^x .
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.
00:36:51.460 --> 00:36:55.010
Now I claim that this is
pretty easy to differentiate.
00:36:55.010 --> 00:37:00.200
Again, it may seem hard, but
it's actually quite easy.
00:37:00.200 --> 00:37:04.590
So maybe somebody
can hazard a guess.
00:37:04.590 --> 00:37:11.530
What's the derivative of x ln a?
00:37:11.530 --> 00:37:14.870
It's just ln a.
00:37:14.870 --> 00:37:18.400
So this is the same thing that I
was talking about before, which
00:37:18.400 --> 00:37:21.420
is if you've got 3x,
and you're taking
00:37:21.420 --> 00:37:24.804
its derivative with respect
to x here, that's just 3.
00:37:24.804 --> 00:37:26.220
That's the kind
of thing you have.
00:37:26.220 --> 00:37:30.110
Again, don't be put off by this
massive piece of junk here.
00:37:30.110 --> 00:37:33.260
It's a constant.
00:37:33.260 --> 00:37:38.220
So again, keep that in mind.
00:37:38.220 --> 00:37:42.460
It comes up regularly in
this kind of question.
00:37:42.460 --> 00:37:46.980
So there's our formula, that the
logarithmic derivative is this.
00:37:46.980 --> 00:37:50.360
But let's just rewrite that.
00:37:50.360 --> 00:37:58.600
That's the same thing as u' / u,
which is (ln u)' = ln a, right?
00:37:58.600 --> 00:38:00.610
So this is our
differentiation formula.
00:38:00.610 --> 00:38:01.800
So here we have u'.
00:38:01.800 --> 00:38:07.460
u' is equal to u times ln a, if
I just multiply through by u.
00:38:07.460 --> 00:38:08.630
And that's what we wanted.
00:38:08.630 --> 00:38:16.660
That's d/dx a^x is equal to
ln a (I'll reverse the order
00:38:16.660 --> 00:38:24.640
of the two, which is
customary) times a^x.
00:38:24.640 --> 00:38:27.390
So this is the way that
logarithmic differentiation
00:38:27.390 --> 00:38:27.890
works.
00:38:27.890 --> 00:38:32.680
It's the same arithmetic
as the previous method,
00:38:32.680 --> 00:38:34.930
but we don't have to
convert to base e.
00:38:34.930 --> 00:38:37.829
We're just keeping
track of the exponents
00:38:37.829 --> 00:38:39.620
and doing differentiation
on the exponents,
00:38:39.620 --> 00:38:44.330
and multiplying
through at the end.
00:38:44.330 --> 00:38:49.660
Okay, so I'm going to do
two trickier examples, which
00:38:49.660 --> 00:39:02.400
illustrate logarithmic
differentiation.
00:39:02.400 --> 00:39:06.490
Again, these could be done
equally well by using base e,
00:39:06.490 --> 00:39:07.730
but I won't do it.
00:39:07.730 --> 00:39:12.120
Method one and method
two always both work.
00:39:12.120 --> 00:39:15.780
So here's a second
example: again this
00:39:15.780 --> 00:39:23.220
is a problem when you
have moving exponents.
00:39:23.220 --> 00:39:25.900
But this time, we're going
to complicate matters
00:39:25.900 --> 00:39:30.520
by having both a moving
exponent and a moving base.
00:39:30.520 --> 00:39:34.710
So we have a function u, which
is, well maybe I'll call it v,
00:39:34.710 --> 00:39:38.640
since we already had a
function u, which is x^x.
00:39:38.640 --> 00:39:41.670
A really complicated
looking function here.
00:39:41.670 --> 00:39:44.490
So again you can handle
this by converting
00:39:44.490 --> 00:39:47.220
to base e, method one.
00:39:47.220 --> 00:39:49.750
But we'll do the logarithmic
differentiation version,
00:39:49.750 --> 00:39:51.110
alright?
00:39:51.110 --> 00:39:59.310
So I take the logs
of both sides.
00:39:59.310 --> 00:40:04.370
And now I differentiate it.
00:40:04.370 --> 00:40:06.200
And now when I
differentiate this here,
00:40:06.200 --> 00:40:07.654
I have to use the product rule.
00:40:07.654 --> 00:40:09.570
This time, instead of
having ln a, a constant,
00:40:09.570 --> 00:40:10.980
I have a variable here.
00:40:10.980 --> 00:40:12.660
So I have two factors.
00:40:12.660 --> 00:40:15.416
I have ln x when I
differentiate with respect to x.
00:40:15.416 --> 00:40:19.410
When I differentiate with
respect to this factor here,
00:40:19.410 --> 00:40:21.800
I get that x times the
derivative of that,
00:40:21.800 --> 00:40:26.910
which is 1/x.
00:40:26.910 --> 00:40:29.160
So, here's my formula.
00:40:29.160 --> 00:40:30.430
Almost finished.
00:40:30.430 --> 00:40:34.701
So I have here v' / v. I'm going
to multiply these two things
00:40:34.701 --> 00:40:35.200
together.
00:40:35.200 --> 00:40:37.741
I'll put it on the other side,
because I don't want to get it
00:40:37.741 --> 00:40:45.340
mixed up with
ln(x+1), the quantity.
00:40:45.340 --> 00:40:47.100
And now I'm almost done.
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 +
00:41:02.110 --> 00:41:04.310
ln x).
00:41:04.310 --> 00:41:13.450
That's it.
00:41:13.450 --> 00:41:31.989
So these two methods always
work for moving exponents.
00:41:31.989 --> 00:41:33.530
So the next thing
that I'd like to do
00:41:33.530 --> 00:41:36.220
is another fairly
tricky example.
00:41:36.220 --> 00:41:45.880
And this one is not strictly
speaking within calculus.
00:41:45.880 --> 00:41:48.800
Although we're going to use the
tools that we just described
00:41:48.800 --> 00:41:52.820
to carry it out, in fact
it will use some calculus
00:41:52.820 --> 00:41:55.990
in the very end.
00:41:55.990 --> 00:41:59.530
And what I'm going to do is
I'm going to evaluate the limit
00:41:59.530 --> 00:42:02.020
as n goes to infinity
of (1 + 1/n)^n.
00:42:11.430 --> 00:42:16.170
So now, the reason why I want
to discuss this is, is it
00:42:16.170 --> 00:42:18.460
turns out to have a
very interesting answer.
00:42:18.460 --> 00:42:23.520
And it's a problem that
you can approach exactly
00:42:23.520 --> 00:42:24.490
by this method.
00:42:24.490 --> 00:42:28.580
And the reason is that
it has a moving exponent.
00:42:28.580 --> 00:42:30.840
The exponent n here is changing.
00:42:30.840 --> 00:42:33.570
And so if you want to keep track
of that, a good way to do that
00:42:33.570 --> 00:42:36.740
is to use logarithms.
00:42:36.740 --> 00:42:38.550
So in order to figure
out this limit,
00:42:38.550 --> 00:42:40.040
we're going to
take the log of it
00:42:40.040 --> 00:42:41.775
and figure out what
the limit of the log
00:42:41.775 --> 00:42:43.280
is, instead of the
log of the limit.
00:42:43.280 --> 00:42:44.990
Those will be the same thing.
00:42:44.990 --> 00:42:48.850
So we're going to take the
natural log of this quantity
00:42:48.850 --> 00:42:56.200
here, and that's n ln(1 + 1/n).
00:43:02.640 --> 00:43:06.370
And now I'm going
to rewrite this
00:43:06.370 --> 00:43:10.960
in a form which will make
it more recognizable,
00:43:10.960 --> 00:43:20.040
so what I'd like to do
is I'm going to write n,
00:43:20.040 --> 00:43:24.730
or maybe I should say it this
way: delta x is equal to 1/n.
00:43:24.730 --> 00:43:29.880
So if n is going to
infinity, then this delta x
00:43:29.880 --> 00:43:33.700
is going to be going to 0.
00:43:33.700 --> 00:43:37.500
So this is more familiar
territory for us in this class,
00:43:37.500 --> 00:43:38.560
anyway.
00:43:38.560 --> 00:43:40.370
So let's rewrite it.
00:43:40.370 --> 00:43:42.980
So here, we have 1 over delta x.
00:43:42.980 --> 00:43:46.680
And then that is multiplied
by ln(1 + delta x).
00:43:50.150 --> 00:43:54.860
So n is the
reciprocal of delta x.
00:43:54.860 --> 00:43:58.330
Now I want to change this
in a very, very minor way.
00:43:58.330 --> 00:44:01.150
I'm going to subtract 0 from it.
00:44:01.150 --> 00:44:02.470
So that's the same thing.
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.
00:44:06.420 --> 00:44:08.260
That's just equal to 0.
00:44:08.260 --> 00:44:10.210
So this is not a
problem, and I'll
00:44:10.210 --> 00:44:14.800
put some parentheses around it.
00:44:14.800 --> 00:44:18.080
Now you're supposed to
recognize, all of a sudden,
00:44:18.080 --> 00:44:21.280
what pattern this fits into.
00:44:21.280 --> 00:44:25.570
This is the thing which we
need to calculate in order
00:44:25.570 --> 00:44:30.810
to calculate the derivative
of the log function.
00:44:30.810 --> 00:44:33.540
So this is, in
the limit as delta
00:44:33.540 --> 00:44:39.300
x goes to 0, equal to
the derivative of ln x.
00:44:39.300 --> 00:44:39.890
Where?
00:44:39.890 --> 00:44:43.940
Well the base point is x=1.
00:44:43.940 --> 00:44:45.710
That's where we're
evaluating it.
00:44:45.710 --> 00:44:46.730
That's the x_0.
00:44:46.730 --> 00:44:49.380
That's the base value.
00:44:49.380 --> 00:44:51.000
So this is the
difference quotient.
00:44:51.000 --> 00:44:52.430
That's exactly what it is.
00:44:52.430 --> 00:44:57.630
And so this by definition
tends to the limit here.
00:44:57.630 --> 00:45:01.470
But we know what the derivative
of the log function is.
00:45:01.470 --> 00:45:03.520
The derivative of the
log function is 1/x.
00:45:09.110 --> 00:45:17.470
So this limit is 1.
00:45:17.470 --> 00:45:18.500
So we got it.
00:45:18.500 --> 00:45:19.730
We got the limit.
00:45:19.730 --> 00:45:22.050
And now we just have
to work backwards
00:45:22.050 --> 00:45:34.010
to figure out what this limit
that we've got over here is.
00:45:34.010 --> 00:45:37.130
So let's do that.
00:45:37.130 --> 00:45:38.260
So let's see here.
00:45:38.260 --> 00:45:40.570
The log approached 1.
00:45:40.570 --> 00:45:45.960
So the limit as n goes to
infinity of (1 + 1/n)^n.
00:45:49.530 --> 00:45:51.880
So sorry, the log of this.
00:45:51.880 --> 00:45:54.320
Yeah, let's write it this way.
00:45:54.320 --> 00:45:57.530
It's the same thing, as
well, the thing that we know
00:45:57.530 --> 00:46:00.610
is the log of this.
00:46:00.610 --> 00:46:04.700
1 plus 1 over n to the n.
00:46:04.700 --> 00:46:06.390
And goes to infinity.
00:46:06.390 --> 00:46:08.190
That's the one that
we just figured out.
00:46:08.190 --> 00:46:11.380
But now this thing is
the exponential of that.
00:46:11.380 --> 00:46:16.550
So it's really e
to this power here.
00:46:16.550 --> 00:46:19.060
So this guy is the
same as the limit
00:46:19.060 --> 00:46:21.799
of the log of the limit of the
thing, which is the same as log
00:46:21.799 --> 00:46:22.340
of the limit.
00:46:22.340 --> 00:46:25.268
The limit of the log and the
log of the limit are the same.
00:46:25.268 --> 00:46:32.260
log lim equals lim log.
00:46:32.260 --> 00:46:33.754
Okay, so I take
the logarithm, then
00:46:33.754 --> 00:46:35.170
I'm going to take
the exponential.
00:46:35.170 --> 00:46:37.800
That just undoes
what I did before.
00:46:37.800 --> 00:46:41.910
And so this limit is
just 1, so this is e^1.
00:46:41.910 --> 00:46:52.000
And so the limit that we
want here is equal to e.
00:46:52.000 --> 00:46:56.790
So I claim that with this
step, we've actually closed
00:46:56.790 --> 00:46:58.270
the loop, finally.
00:46:58.270 --> 00:47:03.620
Because we have an honest
numerical way to calculate e.
00:47:03.620 --> 00:47:04.140
The first.
00:47:04.140 --> 00:47:05.400
There are many such.
00:47:05.400 --> 00:47:07.640
But this one is a perfectly
honest numerical way
00:47:07.640 --> 00:47:08.680
to calculate e.
00:47:08.680 --> 00:47:09.950
We had this thing.
00:47:09.950 --> 00:47:12.170
We didn't know
exactly what it was.
00:47:12.170 --> 00:47:14.872
It was this M(e), there was
M(a), the logarithm, and so on.
00:47:14.872 --> 00:47:15.830
We have all that stuff.
00:47:15.830 --> 00:47:18.920
But we really need to nail
down what this number e is.
00:47:18.920 --> 00:47:20.920
And this is telling
us, if you take
00:47:20.920 --> 00:47:25.620
for example 1 plus 1 over 100
to the 100th power, that's
00:47:25.620 --> 00:47:28.910
going to be a very good,
perfectly decent anyway,
00:47:28.910 --> 00:47:30.730
approximation to e.
00:47:30.730 --> 00:47:36.330
So this is a numerical
approximation,
00:47:36.330 --> 00:47:39.320
which is all we can
ever do with just
00:47:39.320 --> 00:47:42.700
this kind of irrational number.
00:47:42.700 --> 00:47:46.270
And so that closes
the loop, and we now
00:47:46.270 --> 00:47:49.452
have a coherent
family of functions,
00:47:49.452 --> 00:47:51.660
which are actually well
defined and for which we have
00:47:51.660 --> 00:47:54.550
practical methods to calculate.
00:47:54.550 --> 00:47:56.312
Okay, see you next time.