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PROFESSOR: Hi.
00:00:28.340 --> 00:00:31.370
Our subject matter today
concerns that phase of
00:00:31.370 --> 00:00:36.080
calculus known as the
'indefinite integral', or the
00:00:36.080 --> 00:00:38.010
'antiderivative'.
00:00:38.010 --> 00:00:41.260
In terms of the concepts that
we've talked about so far in
00:00:41.260 --> 00:00:45.930
our course, and keeping in mind
that one concept with
00:00:45.930 --> 00:00:50.050
many applications is
educationally more meaningful
00:00:50.050 --> 00:00:54.320
than many concepts each with one
application, I prefer to
00:00:54.320 --> 00:00:59.810
call today's lesson the inverse
derivative, or inverse
00:00:59.810 --> 00:01:01.340
differentiation.
00:01:01.340 --> 00:01:05.530
And again, bring into play a
very well-known feature of our
00:01:05.530 --> 00:01:08.830
course, namely, our discussion
of inverse functions.
00:01:08.830 --> 00:01:12.750
Now, you see, the only problem
that comes up in this context
00:01:12.750 --> 00:01:16.240
is a rather simple one and that
is that all the time that
00:01:16.240 --> 00:01:18.820
we've been taking a derivative,
we may not have
00:01:18.820 --> 00:01:23.010
realized that we were using the
idea of a function in a
00:01:23.010 --> 00:01:24.990
rather different way.
00:01:24.990 --> 00:01:28.730
Namely, noticed that by taking
a derivative, we had a rule
00:01:28.730 --> 00:01:33.620
which told us how, given a
particular function, to assign
00:01:33.620 --> 00:01:37.260
to it a new function called
its 'derivative'.
00:01:37.260 --> 00:01:40.130
In other words, what we could
have done is to have
00:01:40.130 --> 00:01:43.620
visualized a different function
machine, which I will
00:01:43.620 --> 00:01:45.490
call the 'D-machine'.
00:01:45.490 --> 00:01:49.050
Hopefully, the 'D' will suggest
differentiation, where
00:01:49.050 --> 00:01:51.130
the domain of my the machine--
00:01:51.130 --> 00:01:54.410
in other words, the input of
my 'D-machine' will be
00:01:54.410 --> 00:01:59.620
differentiable functions and the
output, you see, the image
00:01:59.620 --> 00:02:03.190
of my 'D-machine' will
be the derivative of
00:02:03.190 --> 00:02:05.290
that particular function.
00:02:05.290 --> 00:02:08.310
So in other words then, if we
want to use our typical
00:02:08.310 --> 00:02:11.500
notation, what we're
saying is that the
00:02:11.500 --> 00:02:13.080
'D-machine' does what?
00:02:13.080 --> 00:02:16.480
Given a differentiable function
as its input, the
00:02:16.480 --> 00:02:19.830
output will be the derivative
'f prime'.
00:02:19.830 --> 00:02:22.710
Now again, we're used to talking
about the derivative
00:02:22.710 --> 00:02:25.080
with respect to a
given variable.
00:02:25.080 --> 00:02:28.950
Unless otherwise specified,
the variable is 'x'.
00:02:28.950 --> 00:02:32.150
And so, perhaps with that in
mind, maybe it would be more
00:02:32.150 --> 00:02:37.310
suggestive if I were to do
something like this to
00:02:37.310 --> 00:02:41.540
indicate, you see, that the
input is 'f of x' and the
00:02:41.540 --> 00:02:44.250
output is 'f prime of x'.
00:02:44.250 --> 00:02:48.230
Now, you may recall also that
long before we started dealing
00:02:48.230 --> 00:02:51.190
with functions of real
variables, we discussed
00:02:51.190 --> 00:02:53.840
functions in general
in terms of circle
00:02:53.840 --> 00:02:55.470
diagrams and the like.
00:02:55.470 --> 00:02:59.020
In this respect, notice that
we can think of our
00:02:59.020 --> 00:03:03.740
'D-machine' as operating on a
particular domain, the domain
00:03:03.740 --> 00:03:04.650
being what?
00:03:04.650 --> 00:03:08.070
The set of all differentiable
functions.
00:03:08.070 --> 00:03:12.600
And the range, or the image,
will also be what?
00:03:12.600 --> 00:03:15.640
All functions which
are derivatives of
00:03:15.640 --> 00:03:17.780
differentiable functions.
00:03:17.780 --> 00:03:20.420
Now, what kind of a
function is 'D'?
00:03:20.420 --> 00:03:22.960
Don't lose track of the
fact that things like
00:03:22.960 --> 00:03:27.800
one-to-oneness and ontoness are
concepts that transcend
00:03:27.800 --> 00:03:29.570
dealing with numbers.
00:03:29.570 --> 00:03:31.440
They apply whenever
we're dealing with
00:03:31.440 --> 00:03:32.850
any kind of a function.
00:03:32.850 --> 00:03:34.590
The idea is something
like this.
00:03:34.590 --> 00:03:38.010
If 'x squared' were to go into
the 'D-machine', the output
00:03:38.010 --> 00:03:40.020
would be '2x'.
00:03:40.020 --> 00:03:43.060
See? 'x squared' is
mapped into '2x'.
00:03:43.060 --> 00:03:46.920
'x cubed' is mapped into
'3 x squared'.
00:03:46.920 --> 00:03:49.230
See, the derivative of 'x cubed'
with respect to 'x' is
00:03:49.230 --> 00:03:50.710
'3 x squared'.
00:03:50.710 --> 00:03:55.590
'x squared plus 7', also
mapped into '2x'.
00:03:55.590 --> 00:03:58.350
In other words, without going
any further, notice that
00:03:58.350 --> 00:04:06.350
whatever 'D' is, 'D'
is not 1 to 1.
00:04:06.350 --> 00:04:09.950
You see, two different functions
can have the same
00:04:09.950 --> 00:04:10.930
derivative.
00:04:10.930 --> 00:04:14.860
In fact, many different
functions can have the same
00:04:14.860 --> 00:04:15.730
derivative.
00:04:15.730 --> 00:04:18.620
For example, once we know that
the derivative of 'x squared'
00:04:18.620 --> 00:04:22.180
is '2x', we certainly know
that the derivative of 'x
00:04:22.180 --> 00:04:26.080
squared' plus any constant
is also '2x'.
00:04:26.080 --> 00:04:29.050
In other words then, we cannot
make an inverse function
00:04:29.050 --> 00:04:31.400
machine as things stand here.
00:04:31.400 --> 00:04:32.530
And why is that?
00:04:32.530 --> 00:04:36.360
To have an inverse function,
what you must be able to do is
00:04:36.360 --> 00:04:39.930
to reverse arrowheads and
still have a function.
00:04:39.930 --> 00:04:44.420
Notice that, in a way, 'D
inverse' would be a little bit
00:04:44.420 --> 00:04:46.320
of a tricky thing to
talk about here.
00:04:46.320 --> 00:04:47.940
But let's pretend that
we could anyway.
00:04:47.940 --> 00:04:50.960
In other words, what would go
wrong if we tried to build a
00:04:50.960 --> 00:04:53.230
'D' inverse machine?
00:04:53.230 --> 00:04:56.340
Well, in terms of the example
that we're talking about, if
00:04:56.340 --> 00:05:01.780
the input to the D inverse
machine happened to be '2x',
00:05:01.780 --> 00:05:04.390
then there would be infinitely
many different outputs that
00:05:04.390 --> 00:05:06.050
we're sure of.
00:05:06.050 --> 00:05:06.880
Namely what?
00:05:06.880 --> 00:05:09.830
Every function of the form
'x squared plus c'
00:05:09.830 --> 00:05:11.410
where 'c' is a constant.
00:05:11.410 --> 00:05:14.450
And what do I mean, we can
be sure of that many?
00:05:14.450 --> 00:05:17.880
You see, all I know going back
to this diagram here is that
00:05:17.880 --> 00:05:21.980
every function of the form 'x
squared plus c' will map into
00:05:21.980 --> 00:05:24.550
'2x' under the derivative
function.
00:05:24.550 --> 00:05:28.170
The other question that comes
up is, how do you know that
00:05:28.170 --> 00:05:32.690
you can find a function whose
derivative is '2x' that comes
00:05:32.690 --> 00:05:35.100
from a function which doesn't
have the form 'x
00:05:35.100 --> 00:05:36.210
squared plus c'?
00:05:36.210 --> 00:05:39.480
How do you know there isn't some
other function 'f of x'
00:05:39.480 --> 00:05:42.580
which has the property that
'f of x' maps into '2x'?
00:05:42.580 --> 00:05:45.640
In other words, the derivative
of 'f of x' is '2x' even
00:05:45.640 --> 00:05:50.040
though 'f of x' does not have
the form 'x squared plus c'.
00:05:50.040 --> 00:05:53.810
And this is exactly where the
mean value theorem comes in to
00:05:53.810 --> 00:05:55.330
help us over here.
00:05:55.330 --> 00:05:59.040
You see, what we're saying here
is, suppose that 'D' of
00:05:59.040 --> 00:06:00.590
'f of x' is '2x'.
00:06:00.590 --> 00:06:03.300
All we know is that when you
run 'f of x' through the
00:06:03.300 --> 00:06:06.970
'D-machine', you wind
up with '2x'.
00:06:06.970 --> 00:06:08.050
What does that mean?
00:06:08.050 --> 00:06:11.010
'f prime of x' is '2x'.
00:06:11.010 --> 00:06:14.700
Well, we know the derivative of
'x squared' is also '2x'.
00:06:14.700 --> 00:06:19.360
Therefore, whatever 'f of x'
is by the corollary to the
00:06:19.360 --> 00:06:22.690
mean value theorem that we've
studied before, since 'f of x'
00:06:22.690 --> 00:06:27.200
and 'x squared' have identical
derivatives, namely, they're
00:06:27.200 --> 00:06:31.930
both '2x', it means that they
must differ by a constant.
00:06:31.930 --> 00:06:35.580
In other words, 'f of x' minus
'x squared' is a constant.
00:06:35.580 --> 00:06:38.700
But to say that 'f of x' minus
'x squared' is a constant is
00:06:38.700 --> 00:06:43.000
the same as saying that 'f of
x' belongs to the family 'x
00:06:43.000 --> 00:06:44.580
squared plus c'.
00:06:44.580 --> 00:06:46.920
In other words, 'f
of x' equals 'x
00:06:46.920 --> 00:06:49.420
squared' plus a constant.
00:06:49.420 --> 00:06:52.280
What the mean value theorem
tells us is not only does
00:06:52.280 --> 00:06:55.430
every function of the form 'x
squared plus c' have its
00:06:55.430 --> 00:06:58.120
derivative equal to '2x' but
that every function whose
00:06:58.120 --> 00:07:01.980
derivative is '2x' has the
form 'x squared plus c'.
00:07:04.560 --> 00:07:06.960
To generalize this, consider
the following.
00:07:06.960 --> 00:07:10.570
Suppose we're given
'f of x', OK?
00:07:10.570 --> 00:07:16.040
And we now say, look, every
time you differentiate a
00:07:16.040 --> 00:07:19.650
function, if you add on a
constant, you don't change
00:07:19.650 --> 00:07:21.200
anything, meaning what?
00:07:21.200 --> 00:07:22.960
The derivative of
a constant is 0.
00:07:22.960 --> 00:07:25.480
In other words, what we're
saying is, if the derivative
00:07:25.480 --> 00:07:30.100
of 'f of x' is 'f prime of x',
any function of the form 'f of
00:07:30.100 --> 00:07:33.720
x' plus 'c' will have
the same derivative.
00:07:33.720 --> 00:07:37.190
And not only that, the only
function whose derivative is
00:07:37.190 --> 00:07:40.560
'f prime of x' must
be one of these.
00:07:40.560 --> 00:07:43.070
That's what this 'e
sub f' stands for.
00:07:43.070 --> 00:07:45.730
You see, what I'm really saying
here is that certainly
00:07:45.730 --> 00:07:49.730
when you change the constant,
you change the function.
00:07:49.730 --> 00:07:52.220
The point is, with respect
to the thing called the
00:07:52.220 --> 00:07:55.670
derivative, you cannot tell
the difference between two
00:07:55.670 --> 00:07:58.830
functions just by looking at
their derivatives if they
00:07:58.830 --> 00:08:00.290
differ by a constant.
00:08:00.290 --> 00:08:03.600
In other words, with respect
to differentiation, two
00:08:03.600 --> 00:08:06.750
functions which differ by a
constant are equivalent and
00:08:06.750 --> 00:08:10.260
that's why I invented this
notation, 'e of f'.
00:08:10.260 --> 00:08:14.020
All I'm saying is that if we
visualize now that the input
00:08:14.020 --> 00:08:17.850
of the 'D-machine' is not
individual functions but whole
00:08:17.850 --> 00:08:19.890
classes of functions--
00:08:19.890 --> 00:08:22.780
in other words, such that they
differ only by a constant,
00:08:22.780 --> 00:08:25.780
then you see there is a
one-to-one correspondence
00:08:25.780 --> 00:08:28.840
between the input
and the output.
00:08:28.840 --> 00:08:30.610
Now that's a subtle point.
00:08:30.610 --> 00:08:33.049
It's a point which I'm sure
most of you will grasp.
00:08:33.049 --> 00:08:38.240
But more importantly, the
important thing is simply that
00:08:38.240 --> 00:08:42.370
once we've seen one function
with a given derivative, in a
00:08:42.370 --> 00:08:44.960
manner of speaking, we've
seen them all.
00:08:44.960 --> 00:08:47.550
In other words, we only have
enough leeway as to fool
00:08:47.550 --> 00:08:48.825
around with an arbitrary
constant.
00:08:52.410 --> 00:08:54.790
So let's generalize again.
00:08:54.790 --> 00:08:59.250
What do we mean by 'D inverse'
of 'f of x'?
00:08:59.250 --> 00:09:00.550
We mean what?
00:09:00.550 --> 00:09:04.830
The set of all function 'g of
x' whose derivative with
00:09:04.830 --> 00:09:07.955
respect to 'x' is the
given 'f of x'.
00:09:07.955 --> 00:09:10.780
See, that's exactly what the
inverse function means.
00:09:10.780 --> 00:09:14.530
And by the way, notice that
this tells me my set
00:09:14.530 --> 00:09:15.430
implicitly.
00:09:15.430 --> 00:09:17.610
Let me put that in parentheses
over here.
00:09:17.610 --> 00:09:22.520
Namely, suppose somebody says,
does 'g of x' belong to this
00:09:22.520 --> 00:09:24.690
particular set called
'D inverse'?
00:09:24.690 --> 00:09:28.880
All I have to do is
differentiate the given 'g'
00:09:28.880 --> 00:09:30.530
and see if I get 'f'.
00:09:30.530 --> 00:09:33.560
If I do, 'G' belongs
to the set.
00:09:33.560 --> 00:09:36.060
If I don't, it doesn't
belong to the set.
00:09:36.060 --> 00:09:39.050
But as we've so often stressed
about inverse functions,
00:09:39.050 --> 00:09:41.520
notice that the test to see
whether something belongs to
00:09:41.520 --> 00:09:45.710
'D inverse', it's sufficient to
know how to differentiate.
00:09:45.710 --> 00:09:53.320
By the way, to summarize what
we did before, to list this
00:09:53.320 --> 00:09:57.310
thing explicitly, notice that
what we're saying is, to find
00:09:57.310 --> 00:10:01.400
the set of all functions whose
derivative is little 'f of x',
00:10:01.400 --> 00:10:03.420
all we have to do is what?
00:10:03.420 --> 00:10:07.030
Find one function, capital
'F', whose derivative is
00:10:07.030 --> 00:10:08.280
little 'f'.
00:10:08.280 --> 00:10:11.380
And then our set, explicitly,
is what?
00:10:11.380 --> 00:10:14.990
The set of all functions of the
form capital 'F of x' plus
00:10:14.990 --> 00:10:18.540
'c' where 'c' is an arbitrary
constant.
00:10:18.540 --> 00:10:20.580
Now, I'm sure this
concept is not
00:10:20.580 --> 00:10:22.230
difficult for you to grasp.
00:10:22.230 --> 00:10:26.140
For those of you who have been
through calculus before, as
00:10:26.140 --> 00:10:29.170
our course is intended to be for
this type of person too,
00:10:29.170 --> 00:10:32.820
you see, you may not be familiar
with the notation,
00:10:32.820 --> 00:10:35.440
the 'D inverse' notation, which
I want to stress here.
00:10:35.440 --> 00:10:39.380
But the concept, I hope, is
clear in its own right.
00:10:39.380 --> 00:10:44.030
Now of course, you see, the
problem that comes up is that
00:10:44.030 --> 00:10:47.840
it's much easier to
differentiate a function than
00:10:47.840 --> 00:10:52.020
it is to be given a function and
then have to try to find
00:10:52.020 --> 00:10:54.690
what you have to differentiate
to get it.
00:10:54.690 --> 00:10:57.310
Without meaning it as
facetiously as it may sound, I
00:10:57.310 --> 00:11:01.120
use this in on many occasions,
I prefer to say is much
00:11:01.120 --> 00:11:04.910
easier, you see, to scramble an
egg than to unscramble one.
00:11:04.910 --> 00:11:07.950
Told what to differentiate,
that's easy enough to do.
00:11:07.950 --> 00:11:10.730
Given the derivative, that
may not be so easy.
00:11:10.730 --> 00:11:13.780
Let's look in terms
of an example.
00:11:13.780 --> 00:11:18.200
But this is, again, a very
nice teachers trick.
00:11:18.200 --> 00:11:20.250
To get the right answer, you
start with the answer and then
00:11:20.250 --> 00:11:21.120
work to the problem.
00:11:21.120 --> 00:11:25.200
I'll start with the function 'h
of x' equals 'x' times the
00:11:25.200 --> 00:11:26.970
'square root of 'x
squared plus 1''.
00:11:26.970 --> 00:11:31.300
Or written more conveniently in
exponential notation, 'x'
00:11:31.300 --> 00:11:34.050
times ''x squared plus
1' to the 1/2'.
00:11:34.050 --> 00:11:35.670
Let me differentiate that.
00:11:35.670 --> 00:11:36.990
Remember, this is a product.
00:11:36.990 --> 00:11:39.430
The derivative of a
product is what?
00:11:39.430 --> 00:11:42.690
It's the first factor which is
'x' times the derivative of
00:11:42.690 --> 00:11:43.560
the second.
00:11:43.560 --> 00:11:44.270
That means what?
00:11:44.270 --> 00:11:48.490
I bring the 1/2 down to a
power 1 less, times the
00:11:48.490 --> 00:11:51.620
derivative of what's inside
with respect to 'x'.
00:11:51.620 --> 00:11:53.110
See, the some old chain
rule again.
00:11:53.110 --> 00:11:54.410
That's '2x'.
00:11:54.410 --> 00:11:55.160
Then what?
00:11:55.160 --> 00:11:59.610
Plus the second factor, 'x
squared plus 1' to the 1/2
00:11:59.610 --> 00:12:01.770
times the derivative of the
first with respect to 'x',
00:12:01.770 --> 00:12:03.500
which is 1.
00:12:03.500 --> 00:12:08.530
At any rate, simplifying,
bringing the minus 1/2 power
00:12:08.530 --> 00:12:12.360
into the denominator, then
putting everything over a
00:12:12.360 --> 00:12:16.580
common denominator, I wind up
with the fact that 'h prime of
00:12:16.580 --> 00:12:21.340
x' is '2 x squared plus 1' over
the 'square root of 'x
00:12:21.340 --> 00:12:22.890
squared plus 1''.
00:12:22.890 --> 00:12:24.870
I hope I haven't made a
careless error here.
00:12:24.870 --> 00:12:27.675
But again, one of the beauties
of the new mathematics is that
00:12:27.675 --> 00:12:30.240
it's the method that's
important, OK?
00:12:30.240 --> 00:12:32.250
Now, at any rate,
what do I know?
00:12:32.250 --> 00:12:35.640
Starting with 'h of x' equaling
'x' times the 'square
00:12:35.640 --> 00:12:39.330
root of 'x squared plus 1'', I
now know that its derivative
00:12:39.330 --> 00:12:43.160
is '2 x squared plus 1' over
the 'square root of 'x
00:12:43.160 --> 00:12:44.480
squared plus 1''.
00:12:44.480 --> 00:12:49.190
Let's write that in terms of
our 'D inverse' notation.
00:12:49.190 --> 00:12:53.960
Namely, what we're saying is
that 'x' times the 'square
00:12:53.960 --> 00:12:57.280
root of 'x squared plus 1''
has the property that its
00:12:57.280 --> 00:13:00.620
derivative is '2 x squared plus
1' over the 'square root
00:13:00.620 --> 00:13:02.180
of 'x squared plus 1''.
00:13:02.180 --> 00:13:05.970
Consequently, every function
which has the property that
00:13:05.970 --> 00:13:09.400
its derivative is '2 x squared
plus 1' over the 'square root
00:13:09.400 --> 00:13:13.520
of 'x squared plus 1'' must
come from this family.
00:13:13.520 --> 00:13:14.440
You see?
00:13:14.440 --> 00:13:16.820
Everything of this form
we'll have its
00:13:16.820 --> 00:13:18.000
derivative equal to this.
00:13:18.000 --> 00:13:21.110
And secondly, any other function
cannot have its
00:13:21.110 --> 00:13:23.530
derivative equal to this by
our corollary to the mean
00:13:23.530 --> 00:13:24.670
value theorem.
00:13:24.670 --> 00:13:27.080
Now you see, what
I'm saying is--
00:13:27.080 --> 00:13:30.270
and here's the beauty of what we
mean by inverse operations
00:13:30.270 --> 00:13:31.330
and the like.
00:13:31.330 --> 00:13:34.310
It's conceivable that you
might not have been
00:13:34.310 --> 00:13:37.380
sophisticated enough at this
stage in the game to have been
00:13:37.380 --> 00:13:40.650
able to deduce this had
we been given this.
00:13:40.650 --> 00:13:44.330
Notice that my cute trick was I
started with this, found out
00:13:44.330 --> 00:13:46.160
what the derivative
was, and then just
00:13:46.160 --> 00:13:47.800
inverted the emphasis.
00:13:47.800 --> 00:13:49.650
A change in emphasis again.
00:13:49.650 --> 00:13:52.500
However, notice the following.
00:13:52.500 --> 00:13:54.810
Suppose you weren't
able to find this.
00:13:54.810 --> 00:13:58.810
And somebody said to you, I
wonder if 'x' times the
00:13:58.810 --> 00:14:01.900
'square root of 'x squared plus
1'' is a function whose
00:14:01.900 --> 00:14:03.990
derivative is equal to this.
00:14:03.990 --> 00:14:06.600
And all I'm saying, without
going through the work again
00:14:06.600 --> 00:14:09.670
because I've already done that,
is to simply observe
00:14:09.670 --> 00:14:13.520
that even if you did not know
this explicit representation,
00:14:13.520 --> 00:14:18.400
by definition, 'D inverse' of '2
x squared plus 1' over the
00:14:18.400 --> 00:14:21.310
'square root of 'x squared
plus 1'' is simply what?
00:14:21.310 --> 00:14:25.530
The set of all functions 'g of
x' such that 'g prime of x' is
00:14:25.530 --> 00:14:28.510
equal to '2 x squared plus 1'
over the 'square root of 'x
00:14:28.510 --> 00:14:29.620
squared plus 1''.
00:14:29.620 --> 00:14:32.090
In other words, given any
function at all, I could
00:14:32.090 --> 00:14:33.210
differentiate it.
00:14:33.210 --> 00:14:35.180
If the derivative came
out to be this,
00:14:35.180 --> 00:14:36.700
then I have a solution.
00:14:36.700 --> 00:14:38.350
It belongs to the
solution set.
00:14:38.350 --> 00:14:39.720
Otherwise, it doesn't.
00:14:39.720 --> 00:14:44.380
But again, notice that's to
solve any 'D inverse' problem,
00:14:44.380 --> 00:14:46.270
it's sufficient to understand a
00:14:46.270 --> 00:14:49.170
corresponding derivative property.
00:14:49.170 --> 00:14:53.060
In fact, maybe now is a good
time to show how we get
00:14:53.060 --> 00:14:57.530
certain recipes for 'D
inverse' type things.
00:14:57.530 --> 00:15:00.490
Let me just write down
a typical one.
00:15:00.490 --> 00:15:05.060
You see, 'D inverse of 'x to the
n'' is 'x to the 'n + 1''
00:15:05.060 --> 00:15:07.770
over 'n + 1' plus a constant.
00:15:07.770 --> 00:15:10.920
And of course, observe that as
soon as you see something like
00:15:10.920 --> 00:15:14.120
this, you have to beware of
'n' equals negative 1.
00:15:14.120 --> 00:15:16.950
Otherwise we have
a 0 denominator.
00:15:16.950 --> 00:15:19.700
Now, a person says, this doesn't
look familiar to me.
00:15:19.700 --> 00:15:23.870
Again, keep in mind what
D inverse means.
00:15:23.870 --> 00:15:28.390
Essentially, to say this is just
a switch in emphasis from
00:15:28.390 --> 00:15:30.020
saying what?
00:15:30.020 --> 00:15:33.990
That if you run the family of
functions 'x to the 'n + 1''
00:15:33.990 --> 00:15:38.670
over 'n + 1' plus a constant
through your 'D-machine', you
00:15:38.670 --> 00:15:39.770
get 'x to the n'.
00:15:39.770 --> 00:15:41.930
Or more familiarly, what?
00:15:41.930 --> 00:15:46.490
The derivative of any member in
this family is 'x to the n'
00:15:46.490 --> 00:15:49.750
provided that 'n' is not
equal to minus 1.
00:15:49.750 --> 00:15:52.760
Now again, this may look a
little bit abstract to you.
00:15:52.760 --> 00:15:56.590
So to avoid this problem,
let's just do a concrete
00:15:56.590 --> 00:15:57.670
illustration.
00:15:57.670 --> 00:15:59.740
Let's pick a particular
value of 'n' and
00:15:59.740 --> 00:16:01.490
work with this thing.
00:16:01.490 --> 00:16:04.430
Let's suppose we're told to
determine 'D inverse'
00:16:04.430 --> 00:16:05.660
of 'x to the 7th'.
00:16:05.660 --> 00:16:07.050
What does that mean?
00:16:07.050 --> 00:16:11.420
What it really means is, let's
find a function whose
00:16:11.420 --> 00:16:13.660
derivative is 'x to the 7th'.
00:16:13.660 --> 00:16:16.150
And why do I say let's
find a function?
00:16:16.150 --> 00:16:19.550
Because once I find a function,
all I have to do is
00:16:19.550 --> 00:16:22.840
tack on arbitrary constants and
the family that I get that
00:16:22.840 --> 00:16:26.460
way is the unique family of
functions which have this
00:16:26.460 --> 00:16:27.870
particular derivative.
00:16:27.870 --> 00:16:29.730
So I play the detective game.
00:16:29.730 --> 00:16:32.680
I know from differential
calculus that if I
00:16:32.680 --> 00:16:36.400
differentiate 'x' to the 8th
power, I'll wind up with the
00:16:36.400 --> 00:16:37.620
exponent 7, at least.
00:16:37.620 --> 00:16:40.950
In other words, the derivative
of 'x' to the 8th power.
00:16:40.950 --> 00:16:45.250
'D of 'x to the 8th'' is
'8 'x to the 7th''.
00:16:45.250 --> 00:16:47.340
Well, what answer did
I want to get?
00:16:47.340 --> 00:16:51.010
I wanted to get 'x to the 7th',
not '8 'x to the 7th''.
00:16:51.010 --> 00:16:53.230
So I fudge this thing
a little bit.
00:16:53.230 --> 00:16:56.220
I say, evidently what I should
have done was to have started
00:16:56.220 --> 00:16:57.890
with 1/8 as much.
00:16:57.890 --> 00:17:01.290
In other words, multiplying
equals by equals, I multiply
00:17:01.290 --> 00:17:05.950
both sides of this equation by
1/8 and I wind up with '1/8 D
00:17:05.950 --> 00:17:08.730
'x to the 8th'' equals
'x to the 7th'.
00:17:08.730 --> 00:17:11.240
And now comes a very
crucial step.
00:17:11.240 --> 00:17:14.089
And let me write that down
because I think it's something
00:17:14.089 --> 00:17:17.329
that we should pay very
close attention to.
00:17:17.329 --> 00:17:20.480
And that is that the derivative
has the property
00:17:20.480 --> 00:17:23.599
that if you want to
differentiate a constant times
00:17:23.599 --> 00:17:28.420
a function, you can take the
constant out and differentiate
00:17:28.420 --> 00:17:29.730
just the function.
00:17:29.730 --> 00:17:32.210
This is a very crucial point
because you see--
00:17:32.210 --> 00:17:35.180
and by the way, notice that in
general, not all functions
00:17:35.180 --> 00:17:37.690
have this property.
00:17:37.690 --> 00:17:42.100
For example, if you're squaring
something, if you
00:17:42.100 --> 00:17:46.810
double the number that's being
squared, the output is 4 times
00:17:46.810 --> 00:17:49.290
as much because twice
something squared
00:17:49.290 --> 00:17:50.320
is 4 times as much.
00:17:50.320 --> 00:17:54.050
In other words, in general,
you do not say that if you
00:17:54.050 --> 00:17:56.840
double the input, you're going
to double the output.
00:17:56.840 --> 00:17:58.620
Not every function has
that property.
00:17:58.620 --> 00:18:01.420
But the function called 'D', the
derivative, does have this
00:18:01.420 --> 00:18:02.620
particular property.
00:18:02.620 --> 00:18:04.600
And you see, with that
in mind, I can
00:18:04.600 --> 00:18:06.460
bring this 1/8 inside.
00:18:06.460 --> 00:18:12.330
This is the key step, that 1/8
times the derivative of 'x to
00:18:12.330 --> 00:18:16.190
the 8th' is a derivative of
''1/8' x to the 8th'.
00:18:16.190 --> 00:18:17.850
That's this key step
over here.
00:18:17.850 --> 00:18:20.900
And now, you see, putting all
this together, I find what?
00:18:20.900 --> 00:18:23.940
That the derivative of
''1/8' x to the 8th'
00:18:23.940 --> 00:18:25.840
is 'x to the 7th'.
00:18:25.840 --> 00:18:28.950
And therefore, since I found one
function whose derivative
00:18:28.950 --> 00:18:32.560
is 'x to the 7th', I have, in
a sense, found them all,
00:18:32.560 --> 00:18:35.830
namely, ''1/8' x to
the 8th' plus 'c'.
00:18:35.830 --> 00:18:39.180
In other words, that's what I
call the equivalent class of
00:18:39.180 --> 00:18:40.270
''1/8' x to the 8th'.
00:18:40.270 --> 00:18:43.030
All the functions that differ
from ''1/8' x to
00:18:43.030 --> 00:18:45.350
the 8th' by a constant.
00:18:45.350 --> 00:18:48.720
And again, to emphasize this
very important point, let me
00:18:48.720 --> 00:18:54.010
again mention, beware of
non-constant factors.
00:18:54.010 --> 00:18:57.180
Let me give you a
for instance.
00:18:57.180 --> 00:19:00.800
Let's suppose I take almost
the same problem.
00:19:00.800 --> 00:19:02.650
And that almost as
a big almost.
00:19:02.650 --> 00:19:05.910
Let's suppose I say, let me
find all functions whose
00:19:05.910 --> 00:19:09.320
derivative, say, is 'x squared
plus 1' to the 7th power.
00:19:09.320 --> 00:19:12.310
In other words, still something
to the 7th power.
00:19:12.310 --> 00:19:14.930
So I argue something like,
well, since whenever I
00:19:14.930 --> 00:19:18.460
differentiate I reduce the
exponent by 1, to wind up with
00:19:18.460 --> 00:19:20.330
a 7th power, maybe
I should have
00:19:20.330 --> 00:19:22.490
started with an 8th power.
00:19:22.490 --> 00:19:24.850
So I say, OK, that's
what I'll do.
00:19:24.850 --> 00:19:26.290
I'll start with an 8th power.
00:19:26.290 --> 00:19:28.910
So I say, OK, what is the
derivative of 'x squared plus
00:19:28.910 --> 00:19:30.490
1' to the 8th power?
00:19:30.490 --> 00:19:33.560
Now notice I know how to
differentiate, hopefully.
00:19:33.560 --> 00:19:36.040
And again, let me make this
point very strongly.
00:19:36.040 --> 00:19:39.280
There is no sense studying
inverse functions if we don't
00:19:39.280 --> 00:19:41.180
know the original
function itself.
00:19:41.180 --> 00:19:43.710
Because the whole purpose of the
inverse function, or the
00:19:43.710 --> 00:19:47.410
whole strategy behind it, is to
reduce it to the original
00:19:47.410 --> 00:19:50.020
function, namely, to switch
the emphasis.
00:19:50.020 --> 00:19:52.510
So at any rate, I differentiate
'x squared plus
00:19:52.510 --> 00:19:53.760
1' to the 8th power.
00:19:53.760 --> 00:19:56.970
I get '8 'x squared plus
1' to the 7th'.
00:19:56.970 --> 00:20:01.410
But now, by the chain rule, I
must remember that this part
00:20:01.410 --> 00:20:04.330
was only the derivative with
respect to 'x squared plus 1'.
00:20:04.330 --> 00:20:05.740
The correction factor is what?
00:20:05.740 --> 00:20:07.960
The derivative of what's inside
with respect to 'x'.
00:20:07.960 --> 00:20:09.000
That's '2x'.
00:20:09.000 --> 00:20:12.180
And so I wind up with that if
I differentiate ''x squared
00:20:12.180 --> 00:20:15.960
plus 1' to the 8th', I get '16x'
times ''x squared plus
00:20:15.960 --> 00:20:17.100
1' to the 7th'.
00:20:17.100 --> 00:20:18.590
Now, how much did
I want to get?
00:20:18.590 --> 00:20:21.980
I wanted to get just ''x squared
plus 1' to the 7th'.
00:20:21.980 --> 00:20:24.990
That put me off by a
factor of '16x'.
00:20:24.990 --> 00:20:28.070
Now I say, OK, I'll
fix that up.
00:20:28.070 --> 00:20:32.200
Namely, I'll divide both sides
by '16x', assuming, of course,
00:20:32.200 --> 00:20:33.430
that 'x' is not 0.
00:20:33.430 --> 00:20:36.450
And this, by the way,
is perfectly valid.
00:20:36.450 --> 00:20:41.360
I can now go from here to here
and say, look, '1 over 16x'
00:20:41.360 --> 00:20:44.720
times the derivative of ''x
squared plus 1' to the 8th'
00:20:44.720 --> 00:20:48.230
with respect to 'x' is ''x
squared plus 1' to the 7th'.
00:20:48.230 --> 00:20:52.470
However, notice that I cannot
take this factor and bring it
00:20:52.470 --> 00:20:53.560
inside here.
00:20:53.560 --> 00:20:57.220
And again, as I so often have
said, also, of course I can
00:20:57.220 --> 00:20:58.100
bring it inside here.
00:20:58.100 --> 00:20:58.860
I just did.
00:20:58.860 --> 00:21:02.260
What I mean is, I don't
get the right answer.
00:21:02.260 --> 00:21:04.650
And what's the best proof that
I don't get the right answer?
00:21:04.650 --> 00:21:06.600
Very, very simple.
00:21:06.600 --> 00:21:13.400
Take this function,
differentiate it, and see if
00:21:13.400 --> 00:21:15.980
you get ''x squared plus
1' to the 7th'.
00:21:15.980 --> 00:21:18.990
You won't, unless you
differentiate incorrectly.
00:21:18.990 --> 00:21:21.520
Don't be like the person who
just differentiates, brings
00:21:21.520 --> 00:21:26.050
the 8 down, replaces this by
1 less, multiplies by a
00:21:26.050 --> 00:21:29.180
derivative of what's inside,
and cancels everything out.
00:21:29.180 --> 00:21:31.930
Notice that the expression
inside the brackets that I've
00:21:31.930 --> 00:21:34.490
just circled is a quotient.
00:21:34.490 --> 00:21:37.060
And the derivative of a quotient
is obtained in a very
00:21:37.060 --> 00:21:37.830
special way.
00:21:37.830 --> 00:21:40.750
The denominator times the
derivative of the numerator
00:21:40.750 --> 00:21:43.700
minus the numerator times the
derivative of the denominator
00:21:43.700 --> 00:21:45.480
over the square of
the denominator.
00:21:45.480 --> 00:21:48.970
And all I'm saying is, you won't
get ''x squared plus 1'
00:21:48.970 --> 00:21:50.750
to the 7th' power
if you do that.
00:21:50.750 --> 00:21:54.820
Again, notice that you do not
have to know the right answer
00:21:54.820 --> 00:21:59.150
in order to see what answer
is wrong, OK?
00:21:59.150 --> 00:22:00.840
So this would be
a wrong answer.
00:22:00.840 --> 00:22:04.350
By the way, this would also be
a wrong answer because we've
00:22:04.350 --> 00:22:08.010
already seen that the derivative
of ''1/8' x squared
00:22:08.010 --> 00:22:10.130
plus 1' to the 8th
power is what?
00:22:10.130 --> 00:22:13.530
You bring the 8 down, which
kills off the 1/8.
00:22:13.530 --> 00:22:16.620
You replace this to a power of
one less, which gives you ''x
00:22:16.620 --> 00:22:18.390
squared plus 1' to the 7th'.
00:22:18.390 --> 00:22:21.960
But the correction factor here
is you must also multiply by a
00:22:21.960 --> 00:22:24.220
derivative of what's inside
with respect to 'x'.
00:22:24.220 --> 00:22:25.780
And that's to '2x'.
00:22:25.780 --> 00:22:27.350
In other words, you do
not get ''x squared
00:22:27.350 --> 00:22:29.780
plus 1' to the 7th'.
00:22:29.780 --> 00:22:32.970
What you do get is what? ''x
squared plus 1' to the 7th'
00:22:32.970 --> 00:22:34.470
times '2x'.
00:22:34.470 --> 00:22:37.540
And the question that may now
come up is, how come this
00:22:37.540 --> 00:22:41.030
worked when you were raising
'x' to the 8th power but it
00:22:41.030 --> 00:22:43.600
didn't work when you were
raising 'x squared plus 1' to
00:22:43.600 --> 00:22:46.050
the 8th power?
00:22:46.050 --> 00:22:49.580
Again, as always in these
cases, the answer is
00:22:49.580 --> 00:22:53.120
immediately available in
terms of derivatives.
00:22:53.120 --> 00:22:56.580
We can talk, in fact, about
the inverse chain rule.
00:22:56.580 --> 00:22:59.470
That when we really
talked about 'D'--
00:22:59.470 --> 00:23:00.880
remember, we mentioned this at
the very beginning of the
00:23:00.880 --> 00:23:04.900
lecture, that the variable
inside the parentheses was the
00:23:04.900 --> 00:23:08.270
one with respect to which you
were differentiating.
00:23:08.270 --> 00:23:16.480
In other words, what we saw was
that if you wanted to get
00:23:16.480 --> 00:23:20.110
something to the 7th power, what
you had to differentiate
00:23:20.110 --> 00:23:24.690
was 1/8 that same something to
the 8th power if you were
00:23:24.690 --> 00:23:27.670
differentiating with respect
to that same variable.
00:23:27.670 --> 00:23:31.110
In other words, what would have
been ''x squared plus 1'
00:23:31.110 --> 00:23:33.520
to the 7th' would
have been what?
00:23:33.520 --> 00:23:38.060
If you would differentiating
not with respect to 'x' but
00:23:38.060 --> 00:23:42.070
with respect to 'x
squared plus 1'.
00:23:42.070 --> 00:23:45.010
See, what we really wanted when
we wrote this down was
00:23:45.010 --> 00:23:47.220
the derivative with
respect to 'x'.
00:23:47.220 --> 00:23:50.920
And even though this notation
may look a little bit strange
00:23:50.920 --> 00:23:55.130
to you, observe that once you
get used to the notation, this
00:23:55.130 --> 00:23:58.160
is just another way of talking
about the chain rule.
00:23:58.160 --> 00:24:01.830
Namely, to find the derivative
of this with respect to 'x',
00:24:01.830 --> 00:24:05.840
you first take the derivative
with respect to 'x squared
00:24:05.840 --> 00:24:10.180
plus 1' and then multiply that
by the derivative of 'x
00:24:10.180 --> 00:24:13.150
squared plus 1' with
respect to 'x'.
00:24:13.150 --> 00:24:16.140
And if we do that, you see, we
get the answer that we've
00:24:16.140 --> 00:24:17.670
talked about before.
00:24:17.670 --> 00:24:21.710
In other words, what we could
say is that the function that
00:24:21.710 --> 00:24:25.150
you have to differentiate to get
''x squared plus 1' to the
00:24:25.150 --> 00:24:31.060
7th' times '2x' is ''1/8' x
squared plus 1' to the 8th'
00:24:31.060 --> 00:24:32.250
plus a constant.
00:24:32.250 --> 00:24:33.540
And how do I know that?
00:24:33.540 --> 00:24:36.830
Well, the way I know that
is simply what?
00:24:36.830 --> 00:24:40.740
In terms of inverse functions,
I started with ''1/8' x
00:24:40.740 --> 00:24:44.540
squared plus 1' to the 8th',
differentiated it and found
00:24:44.540 --> 00:24:48.190
out I got ''x squared plus 1'
to the 7th' times '2x'.
00:24:48.190 --> 00:24:51.050
And so this became the recipe.
00:24:51.050 --> 00:24:54.170
And again, a rather interesting
aside, if you look
00:24:54.170 --> 00:25:04.620
at this, and look at this, it
would appear at first glance
00:25:04.620 --> 00:25:07.810
that the top one should be a
more difficult problem than
00:25:07.810 --> 00:25:08.810
the bottom one.
00:25:08.810 --> 00:25:13.500
The reason being that the input
seems more simple in the
00:25:13.500 --> 00:25:14.200
bottom one.
00:25:14.200 --> 00:25:17.900
Yet, the interesting point is
that the '2x', which seems to
00:25:17.900 --> 00:25:20.770
make this thing more
complicated, is precisely the
00:25:20.770 --> 00:25:24.510
factor you need by the chain
rule to make this thing work.
00:25:24.510 --> 00:25:27.400
Because when you differentiate
'x squared plus 1' to the 8th
00:25:27.400 --> 00:25:30.120
power, you're going to
get a factor of '2x'
00:25:30.120 --> 00:25:31.360
by the chain rule.
00:25:31.360 --> 00:25:34.570
Now again, the main aim of the
lectures is not to take the
00:25:34.570 --> 00:25:37.460
place of the computational
drill supplied in our
00:25:37.460 --> 00:25:39.670
exercises and in the
text but to give
00:25:39.670 --> 00:25:40.810
you sort of an insight.
00:25:40.810 --> 00:25:44.170
And they'll be plenty of drill
on the mechanics of this in
00:25:44.170 --> 00:25:46.300
our exercises on this section.
00:25:46.300 --> 00:25:49.090
Let me just at least continue
on with the
00:25:49.090 --> 00:25:51.000
concept of our recipes.
00:25:51.000 --> 00:25:54.270
For example, here's
another one.
00:25:54.270 --> 00:25:55.920
And this one says what?
00:25:55.920 --> 00:25:59.165
That if you run the sum of two
functions through the 'D
00:25:59.165 --> 00:26:03.160
inverse' machine, the output is
the same as if you sent the
00:26:03.160 --> 00:26:08.130
functions through separately
and then added them up, OK?
00:26:08.130 --> 00:26:11.290
I'll talk about that in
more detail later.
00:26:11.290 --> 00:26:15.070
Again, all I want to see is
that this is the analogous
00:26:15.070 --> 00:26:18.940
result of, again, a beautiful
property of the derivative.
00:26:18.940 --> 00:26:25.870
And that is that the derivative
of a sum is the sum
00:26:25.870 --> 00:26:27.660
of the derivatives.
00:26:27.660 --> 00:26:29.000
And how does that
work over here?
00:26:29.000 --> 00:26:32.440
Again, a very interesting
property throughout advanced
00:26:32.440 --> 00:26:34.920
calculus, linear algebra
and the like.
00:26:34.920 --> 00:26:37.750
These properties are
very, very special.
00:26:37.750 --> 00:26:40.200
And we'll have occasion, as the
course goes on, to talk
00:26:40.200 --> 00:26:41.220
about them more.
00:26:41.220 --> 00:26:43.840
For the time being, rather than
have you get lost in the
00:26:43.840 --> 00:26:47.440
maze of details, let me work
a specific illustration.
00:26:47.440 --> 00:26:50.980
Let's suppose I would like to
find the family of functions
00:26:50.980 --> 00:26:55.410
whose derivative is 'x to the
5th' plus 'x cubed', OK?
00:26:55.410 --> 00:26:57.050
Now, what I'm saying is this.
00:26:57.050 --> 00:27:00.030
By my previous result, I
certainly know how to find the
00:27:00.030 --> 00:27:03.315
function whose derivative is 'x
to the 5th', namely, ''1/6'
00:27:03.315 --> 00:27:06.450
x to the 6th'.
00:27:06.450 --> 00:27:09.160
I also know how to find the
function whose derivative is
00:27:09.160 --> 00:27:13.200
'x cubed', namely, '1/4'
x to the 4th'.
00:27:13.200 --> 00:27:16.810
Putting these two steps together
and say equals added
00:27:16.810 --> 00:27:21.250
to equals are equal, I can
conclude that 'D of ''1/6' x
00:27:21.250 --> 00:27:25.080
to the sixth'' plus 'D of ''1/4'
x to the fourth'' is 'x
00:27:25.080 --> 00:27:27.080
to the 5th' plus 'x cubed'.
00:27:27.080 --> 00:27:31.030
Now, the key step is that since
the derivative of a sum
00:27:31.030 --> 00:27:34.670
is the sum of the derivatives,
I can say that the sum of
00:27:34.670 --> 00:27:37.800
these two derivatives is the
derivative of the sum of the
00:27:37.800 --> 00:27:38.480
two functions.
00:27:38.480 --> 00:27:40.250
Namely, that this is what?
00:27:40.250 --> 00:27:45.470
'D of ''1/6' x to the 6th'' plus
''1/4' x to the 4th', a
00:27:45.470 --> 00:27:48.540
very important power and
property of the derivative.
00:27:48.540 --> 00:27:52.030
Therefore, have I found one
function whose derivative is
00:27:52.030 --> 00:27:53.830
'x to the 5th' plus 'x cubed'?
00:27:53.830 --> 00:27:56.750
The answer is yes. ''1/6'
x to the 6th' plus
00:27:56.750 --> 00:27:58.420
''1/4' x to the 4th'.
00:27:58.420 --> 00:28:02.040
Therefore, what is the family
of all functions whose
00:28:02.040 --> 00:28:04.690
derivative is 'x 5th'
plus 'x cubed'?
00:28:04.690 --> 00:28:08.030
And again, as before, the
answer is, take your one
00:28:08.030 --> 00:28:11.610
solution that you found, tack on
an arbitrary constant, and
00:28:11.610 --> 00:28:14.060
I suppose, technically speaking,
I should put the
00:28:14.060 --> 00:28:17.490
braces in here to indicate
that my solution is an
00:28:17.490 --> 00:28:22.060
infinite set, all belonging
to one family called an
00:28:22.060 --> 00:28:24.590
equivalent set of functions
because they have the same
00:28:24.590 --> 00:28:25.730
derivative.
00:28:25.730 --> 00:28:29.840
Well, let's continue
on and do a little
00:28:29.840 --> 00:28:31.490
bit more harder stuff.
00:28:31.490 --> 00:28:34.370
Remember, we talked about
implicit differentiation.
00:28:34.370 --> 00:28:39.330
Well, is there an analogue
to implicit 'D inverses'?
00:28:39.330 --> 00:28:43.470
You see, notice that in every
problem so far, when I wrote
00:28:43.470 --> 00:28:47.160
'D inverse', I was essentially
telling you explicitly what
00:28:47.160 --> 00:28:49.040
came out of the 'D-machine'.
00:28:49.040 --> 00:28:51.480
Now, suppose I twist the
emphasis a little bit.
00:28:51.480 --> 00:28:54.670
Suppose I tell you, look, I run
a certain function through
00:28:54.670 --> 00:28:55.780
the 'D-machine'.
00:28:55.780 --> 00:28:57.650
In other words, I form
its derivative.
00:28:57.650 --> 00:29:01.300
What the output is, is the
square of the reciprocal of
00:29:01.300 --> 00:29:02.340
the function.
00:29:02.340 --> 00:29:06.940
In other words, if 'g of x'
comes in, '1 over 'g of x''
00:29:06.940 --> 00:29:08.650
squared comes out.
00:29:08.650 --> 00:29:12.050
And now the question is, what
is the function 'g of x'?
00:29:12.050 --> 00:29:14.970
And again, notice that if we
don't know what the right 'g
00:29:14.970 --> 00:29:18.690
of x', is you can certainly test
a given 'g of x' to see
00:29:18.690 --> 00:29:19.580
whether it's right or not.
00:29:19.580 --> 00:29:20.930
Namely, what could you do?
00:29:20.930 --> 00:29:24.860
You differentiate 'g of x', see
what you get, and if what
00:29:24.860 --> 00:29:30.130
you get isn't 1 over the square
of 'g of x', you've got
00:29:30.130 --> 00:29:31.580
the wrong answer.
00:29:31.580 --> 00:29:34.030
But the question that comes up
is, given this type of a
00:29:34.030 --> 00:29:38.000
problem, how do we handle it?
00:29:38.000 --> 00:29:38.200
See?
00:29:38.200 --> 00:29:42.810
In other words, where does the
inverse idea come in here?
00:29:42.810 --> 00:29:46.710
The implicit relationship that
'g of x' is determined by this
00:29:46.710 --> 00:29:48.100
particular property.
00:29:48.100 --> 00:29:51.410
And again, notice how we use
properties of derivatives.
00:29:51.410 --> 00:29:55.370
We're given that 'g prime of x'
is a synonym, identity, for
00:29:55.370 --> 00:29:58.310
'1 over 'g squared of x'.
00:29:58.310 --> 00:30:02.300
We can cross-multiply and we get
'g squared of x' times 'g
00:30:02.300 --> 00:30:05.060
prime of x' is identically
one.
00:30:05.060 --> 00:30:08.060
Now, if you're clever about
this-- and remember, notice
00:30:08.060 --> 00:30:09.630
this very, very importantly.
00:30:09.630 --> 00:30:13.250
For example, ordinary division
is the inverse of ordinary
00:30:13.250 --> 00:30:14.440
multiplication.
00:30:14.440 --> 00:30:17.410
Notice that to be really cute
in division, you have to be
00:30:17.410 --> 00:30:19.320
pretty cute in multiplication.
00:30:19.320 --> 00:30:22.510
Since all you're doing is
changing the emphasis, notice
00:30:22.510 --> 00:30:28.300
that to handle hard problems in
antiderivatives, you have
00:30:28.300 --> 00:30:30.720
to be able to handle tough
derivative problems.
00:30:30.720 --> 00:30:33.780
What I'm driving at is, you look
at something like this
00:30:33.780 --> 00:30:36.850
and begin to wonder, do you
know a function whose
00:30:36.850 --> 00:30:38.630
derivative is this?
00:30:38.630 --> 00:30:39.620
See?
00:30:39.620 --> 00:30:43.600
The idea is, if you're familiar
with your chain rule,
00:30:43.600 --> 00:30:47.740
what is the derivative
of 'g cubed of x'?
00:30:47.740 --> 00:30:50.140
To differentiate 'g cubed',
what do you do?
00:30:50.140 --> 00:30:54.300
You bring the 3 down to a
power 1 less times the
00:30:54.300 --> 00:30:56.500
derivative of 'g of x'
with respect to 'x'.
00:30:56.500 --> 00:30:57.610
That's your chain rule.
00:30:57.610 --> 00:30:58.920
That's 'g prime of x'.
00:30:58.920 --> 00:31:02.140
In other words, if you're clever
enough to see this,
00:31:02.140 --> 00:31:07.950
what you say here is, OK, now
I multiply both sides by 3.
00:31:07.950 --> 00:31:13.100
The left hand side is just the
derivative of 'g cubed of x'.
00:31:13.100 --> 00:31:17.720
The right hand side is the
derivative off '3x'.
00:31:17.720 --> 00:31:22.210
Therefore, whatever g of x
is, its cube has the same
00:31:22.210 --> 00:31:25.750
derivative with respect
to 'x' as '3x'.
00:31:25.750 --> 00:31:28.600
And we've already learned that
if two functions have
00:31:28.600 --> 00:31:30.520
identical derivatives,
they differ
00:31:30.520 --> 00:31:32.730
by, at most, a constant.
00:31:32.730 --> 00:31:38.120
Consequently, 'g cubed
of x' must equal
00:31:38.120 --> 00:31:40.590
'3x' plus some constant.
00:31:40.590 --> 00:31:47.880
In other words, 'g of x' is the
cube root of '3x plus c'.
00:31:47.880 --> 00:31:51.190
Now, time is running short in
terms of other things that I
00:31:51.190 --> 00:31:52.990
want to teach you in
today's lesson.
00:31:52.990 --> 00:31:56.170
Let me leave this, then,
just for you to check.
00:31:56.170 --> 00:32:01.710
Simply differentiate the cube
root of '3x plus c'.
00:32:01.710 --> 00:32:05.420
And make sure that when you get
that derivative, it does
00:32:05.420 --> 00:32:08.770
turn out to be '1 over
'g of x' squared'.
00:32:08.770 --> 00:32:11.310
As I say, it's a straightforward
demonstration.
00:32:11.310 --> 00:32:12.940
I leave the details to you.
00:32:12.940 --> 00:32:15.980
But the point that's really
important is that whenever you
00:32:15.980 --> 00:32:17.060
do get an answer--
00:32:17.060 --> 00:32:18.720
the hard part is to
get the answer.
00:32:18.720 --> 00:32:22.420
Whenever you do get the answer,
you can check by means
00:32:22.420 --> 00:32:25.260
of just taking a derivative.
00:32:25.260 --> 00:32:29.090
Now, with all of this talk about
'D inverse' in mind, let
00:32:29.090 --> 00:32:32.930
me now go back to the more
traditional notation, the
00:32:32.930 --> 00:32:36.510
notation that you'll find in
most textbooks, the notation
00:32:36.510 --> 00:32:39.720
that, as I say, if you've had
calculus before, most likely,
00:32:39.720 --> 00:32:41.930
you're more familiar with.
00:32:41.930 --> 00:32:46.410
And that is the following, that
when we write 'D inverse'
00:32:46.410 --> 00:32:50.440
of 'f of x', the average
textbook writes
00:32:50.440 --> 00:32:51.880
a symbol like this.
00:32:51.880 --> 00:32:54.740
It's called the 'integral'
of 'f of x'.
00:32:54.740 --> 00:32:59.690
I will have later lectures to
bemoan this choice of notation
00:32:59.690 --> 00:33:00.740
from a different
point of view.
00:33:00.740 --> 00:33:03.410
But for the time being, all
we're saying is, instead of
00:33:03.410 --> 00:33:04.630
writing 'D inverse'--
00:33:04.630 --> 00:33:06.140
again, what's in the name?
00:33:06.140 --> 00:33:09.150
Just use this particular
notation.
00:33:09.150 --> 00:33:12.830
That when you see this
particular thing, perhaps read
00:33:12.830 --> 00:33:14.210
this as what?
00:33:14.210 --> 00:33:18.370
That this particular symbol is a
code to tell you to find all
00:33:18.370 --> 00:33:21.490
functions whose derivative
is 'f of x'.
00:33:21.490 --> 00:33:24.610
And by the way, to summarize
the results that we've
00:33:24.610 --> 00:33:28.980
obtained so far, let me just
rewrite some of these basic
00:33:28.980 --> 00:33:33.240
results in terms of the more
traditional notation.
00:33:33.240 --> 00:33:35.520
When you write this, this is
called the 'indefinite
00:33:35.520 --> 00:33:37.780
integral' and what we're saying
is the indefinite
00:33:37.780 --> 00:33:41.860
integral of ''x to the n' dx' is
'x to the 'n + 1'' over 'n
00:33:41.860 --> 00:33:47.440
+ 1' plus a constant when it
is not equal to minus 1.
00:33:47.440 --> 00:33:51.180
The integral of a sum is the
sum of the integrals.
00:33:51.180 --> 00:33:55.440
And the integral of a constant
times a function is a constant
00:33:55.440 --> 00:33:56.680
times the integral.
00:33:56.680 --> 00:33:58.330
All this says is what?
00:33:58.330 --> 00:34:02.880
That this is a consequent of the
fact that the derivative
00:34:02.880 --> 00:34:04.890
of the sum is the sum
of the derivatives.
00:34:04.890 --> 00:34:08.090
This is a consequent of the fact
that the derivative of a
00:34:08.090 --> 00:34:11.000
constant times a function is
the constant times the
00:34:11.000 --> 00:34:12.590
derivative of the function.
00:34:12.590 --> 00:34:15.870
And again, all this is, same
thing we were talking about
00:34:15.870 --> 00:34:20.739
before only with the 'D inverse'
notation replaced by
00:34:20.739 --> 00:34:25.300
the more common indefinite
integral.
00:34:25.300 --> 00:34:28.179
Now, we come to one more problem
which will finish us
00:34:28.179 --> 00:34:31.800
up for the day, once we get
through talking about it.
00:34:31.800 --> 00:34:32.820
And that's this.
00:34:32.820 --> 00:34:37.110
If all this means is find the
functions whose derivative is
00:34:37.110 --> 00:34:41.239
'f of x', why write this
notation here?
00:34:41.239 --> 00:34:44.000
Why couldn't we have just
written, for example, the
00:34:44.000 --> 00:34:47.739
so-called integral sign with
a little 'x' underneath?
00:34:47.739 --> 00:34:51.730
You see, we've talked about
misleading notations before.
00:34:51.730 --> 00:34:54.460
You see, in terms of our
differential notation, when
00:34:54.460 --> 00:34:58.510
you see ''f of x' dx', you have
every right to think of a
00:34:58.510 --> 00:35:00.010
differential.
00:35:00.010 --> 00:35:04.380
Now, in all fairness, the
chances are that this notation
00:35:04.380 --> 00:35:06.710
would not have been invented
if there weren't some
00:35:06.710 --> 00:35:10.490
connection between derivatives
and differentials.
00:35:10.490 --> 00:35:12.570
So let me mention this point.
00:35:12.570 --> 00:35:15.240
Going back as if we were
starting the lecture all over
00:35:15.240 --> 00:35:18.900
again, could I have invented a
different machine, which I'll
00:35:18.900 --> 00:35:21.910
call the 'script D- machine'?
00:35:21.910 --> 00:35:24.410
In other words, I don't want
to call it the same
00:35:24.410 --> 00:35:27.230
'D-machine' as before because
now it's going to have a
00:35:27.230 --> 00:35:28.660
different set of outputs.
00:35:28.660 --> 00:35:30.530
You see, now the input
will still be
00:35:30.530 --> 00:35:32.100
differentiable functions.
00:35:32.100 --> 00:35:34.780
But the output, instead of being
the derivative of the
00:35:34.780 --> 00:35:38.440
function, will be the
differential of the function.
00:35:38.440 --> 00:35:41.420
For example, before,
we said what?
00:35:41.420 --> 00:35:45.320
If 'x squared' goes in, the
output would be '2x'.
00:35:45.320 --> 00:35:50.420
With the script 'D-machine', the
output would be '2x dx'.
00:35:50.420 --> 00:35:52.950
Now, even though these machines
are different because
00:35:52.950 --> 00:35:55.710
one machine gives you a
differential as an output and
00:35:55.710 --> 00:35:57.900
the other gives you a derivative
as an output,
00:35:57.900 --> 00:35:59.410
notice that they
are equivalent.
00:35:59.410 --> 00:36:04.980
Namely, knowing the
differential, we can pin down
00:36:04.980 --> 00:36:08.390
the function the same as when
we knew the derivative.
00:36:08.390 --> 00:36:10.630
Now, the question that comes up
is, why should we use this
00:36:10.630 --> 00:36:12.440
particular type of notation?
00:36:12.440 --> 00:36:16.410
And to use the examples that are
most prevalent in the text
00:36:16.410 --> 00:36:19.510
and also in our notes, let me
give you the same problem that
00:36:19.510 --> 00:36:21.350
we've done before,
but now from a
00:36:21.350 --> 00:36:22.850
different point of view.
00:36:22.850 --> 00:36:26.210
Suppose we're still given the
problem 'g prime of x' equals
00:36:26.210 --> 00:36:28.480
'1 over 'g of x' squared'.
00:36:28.480 --> 00:36:31.670
We say, OK, let 'y'
equal 'g of x'.
00:36:31.670 --> 00:36:35.270
When we do that, this problem
now translates into what?
00:36:35.270 --> 00:36:40.480
'dy dx' equals '1 over
'y squared''.
00:36:40.480 --> 00:36:44.190
Now, if we allow ourselves to
use the differential notation,
00:36:44.190 --> 00:36:47.810
which we justified in previous
lectures, this says what?
00:36:47.810 --> 00:36:52.000
Cross-multiplying ''y squared'
dy' equals 'dx'.
00:36:52.000 --> 00:36:56.050
Notice, by the way, that 'y' is
implicitly a differentiable
00:36:56.050 --> 00:36:57.840
function of 'x'.
00:36:57.840 --> 00:37:00.780
So what we're saying over here
is, look, here are two
00:37:00.780 --> 00:37:05.850
functions which have the same
differential, therefore, if we
00:37:05.850 --> 00:37:08.630
integrate them, they differ
by a constant.
00:37:08.630 --> 00:37:13.520
Well, to mimic what we were
doing before, let's just say--
00:37:13.520 --> 00:37:14.530
let's see, this, I
think, I have a
00:37:14.530 --> 00:37:15.070
little bit twisted here.
00:37:15.070 --> 00:37:19.300
If ''y squared' dy' equals
'3x', '3 'y squared' dy'
00:37:19.300 --> 00:37:20.900
equals '3dx'.
00:37:20.900 --> 00:37:24.200
If the differentials are equal,
then the functions
00:37:24.200 --> 00:37:25.540
differ by a constant.
00:37:25.540 --> 00:37:28.730
But 'y cubed' is the function
whose differential is '3 'y
00:37:28.730 --> 00:37:31.240
squared' dy'.
00:37:31.240 --> 00:37:34.840
'3x' is the function whose
differential is '3dx'.
00:37:34.840 --> 00:37:40.590
And we wind up with 'y cubed'
equals '3x + c', or, 'y'
00:37:40.590 --> 00:37:45.710
equals the 'cube root
of '3x + c''.
00:37:45.710 --> 00:37:49.120
By the way, let me just pull
this board down so that we can
00:37:49.120 --> 00:37:51.440
make a little bit of
a comparison here.
00:37:51.440 --> 00:37:55.330
You see, notice that in using
this differential notation,
00:37:55.330 --> 00:37:58.770
which allowed us to use some
nice algebraic devices and
00:37:58.770 --> 00:38:01.870
didn't seem quite as difficult
for us to recognize certain
00:38:01.870 --> 00:38:04.820
things, I hope you notice
that there seems to be a
00:38:04.820 --> 00:38:09.200
correspondence between the steps
that we had here and the
00:38:09.200 --> 00:38:11.690
steps that took place
over here.
00:38:11.690 --> 00:38:15.250
In other words, having done at
the so-called rigorous way,
00:38:15.250 --> 00:38:21.620
notice that differentials
give us a very nice,
00:38:21.620 --> 00:38:26.140
intuitively-simpler technique
for solving certain types of
00:38:26.140 --> 00:38:30.720
problems when we have implicit
relationships
00:38:30.720 --> 00:38:33.080
between 'y' and 'x'.
00:38:33.080 --> 00:38:35.540
And so, just to illustrate
this in terms of one more
00:38:35.540 --> 00:38:38.710
problem, let's take one that's
fairly geometric.
00:38:38.710 --> 00:38:41.880
Let's suppose we're given
the following problem.
00:38:41.880 --> 00:38:45.790
'dy dx' is 'minus x' over 'y'
and we know that when 'x'
00:38:45.790 --> 00:38:48.570
equals 3, 'y' equals 4.
00:38:48.570 --> 00:38:51.810
Using the language of
differentials, what we would
00:38:51.810 --> 00:38:53.610
do over here is we would
cross-multiply.
00:38:56.890 --> 00:38:58.040
Whatever you want to do.
00:38:58.040 --> 00:39:03.370
we recognize that, to get '2y',
you must differentiate
00:39:03.370 --> 00:39:04.100
'y squared'.
00:39:04.100 --> 00:39:05.990
This is a step you don't
really need.
00:39:05.990 --> 00:39:07.230
I don't care how you
want to do this.
00:39:07.230 --> 00:39:10.550
All I'm saying is that from this
step, I can get to here.
00:39:10.550 --> 00:39:15.390
Recognizing that y squared has
'2y dy' as its differential
00:39:15.390 --> 00:39:18.550
and that ''minus 'x squared''
has 'minus 2x dx' as its
00:39:18.550 --> 00:39:22.520
differential, I know that 'y
squared' is equal to ''minus
00:39:22.520 --> 00:39:24.260
'x squared'' plus a constant.
00:39:24.260 --> 00:39:25.600
I transpose.
00:39:25.600 --> 00:39:28.800
I get that 'x squared' plus
'y squared' is a constant.
00:39:28.800 --> 00:39:32.380
Knowing that when 'x' equals
3, 'y' equals 4, I can
00:39:32.380 --> 00:39:37.030
determine that the constant must
be 25 when 'x' equals 3
00:39:37.030 --> 00:39:38.250
and 'y' is 4.
00:39:38.250 --> 00:39:41.870
But since it's a constant, if
it's 25 when 'x' equals 3 and
00:39:41.870 --> 00:39:46.050
'y' equals 4, it's
25 everyplace.
00:39:46.050 --> 00:39:48.780
And again, if you want a
geometric interpretation of
00:39:48.780 --> 00:39:53.550
this, the circle centered at the
origin with radius equal
00:39:53.550 --> 00:39:59.550
to 5 is the only curve in the
whole world whose derivative
00:39:59.550 --> 00:40:04.070
at any given point is the
negative of the x-coordinate
00:40:04.070 --> 00:40:07.050
over the y-coordinate
and passes through
00:40:07.050 --> 00:40:09.140
the point (3 , 4).
00:40:09.140 --> 00:40:11.690
And if you want to see that
geometrically, let me just
00:40:11.690 --> 00:40:13.300
take a second here.
00:40:13.300 --> 00:40:19.160
You see, notice that any point
on the circle (x , y), notice
00:40:19.160 --> 00:40:22.560
that the tangent line to the
curve, which has slope 'dy
00:40:22.560 --> 00:40:24.130
dx', is what?
00:40:24.130 --> 00:40:28.370
Perpendicular to the
radius, again.
00:40:28.370 --> 00:40:31.610
The radius has slope
'y' over 'x'.
00:40:31.610 --> 00:40:32.940
And to be perpendicular--
00:40:32.940 --> 00:40:35.000
if two lines are perpendicular,
their slopes
00:40:35.000 --> 00:40:37.610
are negative reciprocals.
00:40:37.610 --> 00:40:40.200
This is this particular
problem.
00:40:40.200 --> 00:40:42.580
And now, all I wanted to show
you is that if you're nervous
00:40:42.580 --> 00:40:45.880
about differentials and you
don't like to use them, notice
00:40:45.880 --> 00:40:48.150
that the same problem could
have been stated without
00:40:48.150 --> 00:40:49.280
differentials.
00:40:49.280 --> 00:40:51.600
Namely, we are thinking of a
function which we'll call 'g
00:40:51.600 --> 00:40:55.310
of x' such that the derivative
of 'g of x' is 'minus
00:40:55.310 --> 00:40:57.370
x' over 'g of x'.
00:40:57.370 --> 00:41:00.670
In other words, if the input of
the 'D-machine' is 'g', the
00:41:00.670 --> 00:41:03.210
output is 'minus x' over 'g'.
00:41:03.210 --> 00:41:06.990
And we also know that when 'x'
equals 3, the output is 4.
00:41:06.990 --> 00:41:09.330
So to 'g' of 3 equals 4.
00:41:09.330 --> 00:41:12.800
And without going through the
great details here, let's just
00:41:12.800 --> 00:41:13.420
notice that we could
00:41:13.420 --> 00:41:16.260
cross-multiply the same as before.
00:41:16.260 --> 00:41:20.580
We could multiply both sides
by 2, the same as before.
00:41:20.580 --> 00:41:23.500
We could recognize that the
left-hand side was the
00:41:23.500 --> 00:41:26.770
derivative of ''g of x'
squared', that the right-hand
00:41:26.770 --> 00:41:29.350
side is the derivative of
'minus 'x squared''.
00:41:29.350 --> 00:41:32.830
Therefore, since they have the
same identical derivatives,
00:41:32.830 --> 00:41:35.270
they must differ
by a constant.
00:41:35.270 --> 00:41:37.070
OK.
00:41:37.070 --> 00:41:41.190
We actually have plus or minus
here, but notice the fact that
00:41:41.190 --> 00:41:45.450
when the input is 3, the output
is 4 means that we're
00:41:45.450 --> 00:41:48.820
on the positive branch of
the curve, et cetera.
00:41:48.820 --> 00:41:52.910
I say et cetera not because
these points aren't important
00:41:52.910 --> 00:41:56.410
but because every point that
comes up now has already been
00:41:56.410 --> 00:42:00.640
discussed under the heading
of differential calculus.
00:42:00.640 --> 00:42:04.370
In other words, that the inverse
of differentiation can
00:42:04.370 --> 00:42:07.490
be handled very, very neatly
just by knowing
00:42:07.490 --> 00:42:10.190
differentiation with a
switch in emphasis.
00:42:10.190 --> 00:42:12.580
Why do we want to know this
particular topic?
00:42:12.580 --> 00:42:16.650
Well, because in many cases, to
look at it geometrically,
00:42:16.650 --> 00:42:19.280
in the past, we were
given the curve.
00:42:19.280 --> 00:42:21.310
We wanted to find out
what the slope was.
00:42:21.310 --> 00:42:24.700
In many physical applications,
we are, in a sense, told what
00:42:24.700 --> 00:42:27.700
the slope is and have to figure
out what the curve is.
00:42:27.700 --> 00:42:29.060
Hence the motivation.
00:42:29.060 --> 00:42:32.440
Well, at any rate, there'll be
more about this in the text
00:42:32.440 --> 00:42:33.790
and in our exercises.
00:42:33.790 --> 00:42:37.890
So until next time, good bye.
00:42:37.890 --> 00:42:40.890
MALE SPEAKER: Funding for the
publication of this video was
00:42:40.890 --> 00:42:45.610
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:42:45.610 --> 00:42:49.780
Help OCW continue to provide
free and open access to MIT
00:42:49.780 --> 00:42:53.980
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