WEBVTT
00:00:00.040 --> 00:00:01.940
NARRATOR: The following content
is provided under a
00:00:01.940 --> 00:00:03.690
Creative Commons license.
00:00:03.690 --> 00:00:06.630
Your support will help MIT
OpenCourseWare continue to
00:00:06.630 --> 00:00:09.990
offer high-quality educational
resources for free.
00:00:09.990 --> 00:00:12.830
To make a donation or view
additional materials from
00:00:12.830 --> 00:00:16.760
hundreds of MIT courses, visit
MIT OpenCourseWare at
00:00:16.760 --> 00:00:18.010
ocw.mit.edu.
00:00:31.700 --> 00:00:32.710
PROFESSOR: Hi.
00:00:32.710 --> 00:00:36.890
Our lecture today is entitled
Implicit Differentiation.
00:00:36.890 --> 00:00:40.440
And aside from other
considerations, this lecture
00:00:40.440 --> 00:00:43.200
bares a very strong relationship
to many of the
00:00:43.200 --> 00:00:47.460
comments that we've made about
single-valued functions and
00:00:47.460 --> 00:00:49.190
one-to-one functions.
00:00:49.190 --> 00:00:53.020
Let's get right into the topic
by talking about one
00:00:53.020 --> 00:00:56.430
particular phase that somehow or
other we may have taken for
00:00:56.430 --> 00:01:01.610
granted, but which really is a
lot more sophisticated than we
00:01:01.610 --> 00:01:03.750
really imagine at
first glance.
00:01:03.750 --> 00:01:08.330
Let's look, for example, at
an expression such as 'y'
00:01:08.330 --> 00:01:09.820
equals 'f of x'.
00:01:09.820 --> 00:01:12.810
A rather harmless expression,
we write it down quite
00:01:12.810 --> 00:01:15.270
frequently, and we say
things like what?
00:01:15.270 --> 00:01:19.470
Given that 'y' equals 'f
of x', find 'dy dx'.
00:01:22.030 --> 00:01:24.840
And you see the point that we've
made here is we have
00:01:24.840 --> 00:01:28.360
assumed that 'y' can
be solved for
00:01:28.360 --> 00:01:30.710
explicitly in terms of 'x'.
00:01:30.710 --> 00:01:33.130
In other words, in terms of
our function machine, we
00:01:33.130 --> 00:01:35.040
visualize an 'f' machine.
00:01:35.040 --> 00:01:39.480
'x' goes in as the input, 'y'
comes out as the output, and
00:01:39.480 --> 00:01:41.180
everything is fine.
00:01:41.180 --> 00:01:44.200
Now, the interesting point
is simply this.
00:01:44.200 --> 00:01:45.810
Let's look at a slightly more
00:01:45.810 --> 00:01:48.110
complicated algebraic equation.
00:01:48.110 --> 00:01:52.100
For example, 'x to the eighth'
plus ''x to the sixth' 'y to
00:01:52.100 --> 00:01:56.460
the fourth'' plus 'y to
the sixth' equals 3.
00:01:56.460 --> 00:02:01.830
Now, in this particular example,
notice that 'x' and
00:02:01.830 --> 00:02:05.190
'y' are not given at random.
00:02:05.190 --> 00:02:09.370
For example, if I say let me
pick 'x' to be 2 and 'y' to be
00:02:09.370 --> 00:02:14.620
2, and I put 2 in for 'x' and
2 in for 'y', just look at
00:02:14.620 --> 00:02:17.040
what happens to the left-hand
side here.
00:02:17.040 --> 00:02:19.280
It makes, in particular,
3 a pretty big number.
00:02:19.280 --> 00:02:23.950
In other words, when 'x' is 2
and 'y' is 2, this equation--
00:02:23.950 --> 00:02:25.930
and we'll talk about the meaning
of the equation more
00:02:25.930 --> 00:02:28.460
in just a few minutes, but
this equation does not
00:02:28.460 --> 00:02:30.230
balance, so to speak.
00:02:30.230 --> 00:02:33.750
In other words, while we cannot
pick 'x' at random for
00:02:33.750 --> 00:02:38.440
a given value of 'y', once 'x'
is given, notice that we wind
00:02:38.440 --> 00:02:42.100
up with a sixth degree
polynomial equation in 'y',
00:02:42.100 --> 00:02:45.290
which determines at most
six values of 'y'.
00:02:45.290 --> 00:02:48.980
And if we're given a value of
'y' as a fixed number, we wind
00:02:48.980 --> 00:02:52.490
up with an eighth degree
polynomial equation in 'x',
00:02:52.490 --> 00:02:53.880
which means that what?
00:02:53.880 --> 00:02:56.650
'x' is no longer
random either.
00:02:56.650 --> 00:03:00.800
The point is it is very, very
difficult, if not impossible,
00:03:00.800 --> 00:03:05.230
to try to solve this particular
equation for 'y'
00:03:05.230 --> 00:03:07.790
explicitly in terms
of 'x' or for 'x'
00:03:07.790 --> 00:03:10.330
explicitly in terms of 'y'.
00:03:10.330 --> 00:03:13.590
And the question is
how do we tackle
00:03:13.590 --> 00:03:15.540
something along these lines?
00:03:15.540 --> 00:03:19.290
And we make a very
tacit assumption.
00:03:19.290 --> 00:03:21.490
And by the way, you'll notice
that when you read this
00:03:21.490 --> 00:03:25.430
section in the text, there are
several remarks saying that
00:03:25.430 --> 00:03:29.590
many of the proofs are beyond
the scope of the textbook at
00:03:29.590 --> 00:03:30.970
this particular point.
00:03:30.970 --> 00:03:33.145
And they'll say we'll talk about
this later, and there
00:03:33.145 --> 00:03:35.890
are references to more
advanced textbooks.
00:03:35.890 --> 00:03:37.520
The point is that from a
more rigorous point of
00:03:37.520 --> 00:03:39.740
view, this is true.
00:03:39.740 --> 00:03:43.400
However, from a geometric
intuitive point of view, it's
00:03:43.400 --> 00:03:46.660
quite easy to see what's really
going on here, as we
00:03:46.660 --> 00:03:48.470
shall do with as the
lecture progresses.
00:03:48.470 --> 00:03:51.280
But for the time being, let's
review what that tacit
00:03:51.280 --> 00:03:52.400
assumption is.
00:03:52.400 --> 00:03:55.930
Essentially what we say is let's
assume that 'y' is a
00:03:55.930 --> 00:03:57.720
particular function of 'x'.
00:03:57.720 --> 00:04:00.060
What particular function
of 'x' is it?
00:04:00.060 --> 00:04:05.730
Well, it's that function of 'x'
such that when you replace
00:04:05.730 --> 00:04:10.450
'y' by that function of 'x',
this becomes an identity.
00:04:10.450 --> 00:04:12.050
And this is what I want
to talk about next.
00:04:12.050 --> 00:04:15.820
You see, assume that 'y' is that
function of 'x' such that
00:04:15.820 --> 00:04:20.180
when you replace 'y' in terms of
'x', 'x to the eighth' plus
00:04:20.180 --> 00:04:23.420
''x to the sixth' 'y the
fourth'' plus 'y to the sixth'
00:04:23.420 --> 00:04:24.980
is identically 3.
00:04:24.980 --> 00:04:29.120
And notice the use of the three
lines here to indicate
00:04:29.120 --> 00:04:32.520
the identity as opposed
to the two lines to
00:04:32.520 --> 00:04:34.720
indicate the equality.
00:04:34.720 --> 00:04:38.070
Now, obviously, there's a more
important difference than to
00:04:38.070 --> 00:04:40.640
say that one is indicated
by three lines and
00:04:40.640 --> 00:04:41.770
the other by two.
00:04:41.770 --> 00:04:46.030
Let me take a few moments to
digress on a very important
00:04:46.030 --> 00:04:49.600
topic, namely, the difference
between what we call an
00:04:49.600 --> 00:04:53.860
equation, or perhaps more
appropriately, a conditional
00:04:53.860 --> 00:04:58.800
equality, and that which we call
an absolute equality or
00:04:58.800 --> 00:04:59.950
an identity.
00:04:59.950 --> 00:05:03.920
For example, let's take a look
at an expression such as 'x
00:05:03.920 --> 00:05:06.310
squared' equals 4.
00:05:06.310 --> 00:05:09.290
You say, ah, these two things
are equals, and if I do the
00:05:09.290 --> 00:05:11.800
same thing to equals,
I get equals.
00:05:11.800 --> 00:05:14.630
So the fellow says I think I'll
differentiate both sides
00:05:14.630 --> 00:05:15.970
of this equality.
00:05:15.970 --> 00:05:19.540
If he differentiates 'x
squared', he gets '2x', and if
00:05:19.540 --> 00:05:22.260
he differentiates
4, he gets 0.
00:05:22.260 --> 00:05:24.760
Now notice that if you solve
the equation 'x squared'
00:05:24.760 --> 00:05:27.180
equals 4, you get what?
00:05:27.180 --> 00:05:30.790
'x' equals minus 2, or 2.
00:05:30.790 --> 00:05:33.600
On the other hand, if you solve
the equation '2x' equals
00:05:33.600 --> 00:05:37.490
0, you get x equals 0, and
there seems to be no
00:05:37.490 --> 00:05:40.510
correlation between these two.
00:05:40.510 --> 00:05:44.990
Now, you see the answer
to this thing is this.
00:05:44.990 --> 00:05:47.660
'x squared' is not a synonym.
00:05:47.660 --> 00:05:51.310
'x squared' is not another
way of saying 4.
00:05:51.310 --> 00:05:54.720
To use the language of the new
mathematics, the solution set
00:05:54.720 --> 00:05:58.070
for 'x squared' equals
4 is what?
00:05:58.070 --> 00:06:03.100
The set of all 'x' such that
'x squared' equals 4.
00:06:03.100 --> 00:06:05.660
That's another way
of saying what?
00:06:05.660 --> 00:06:10.330
The set whose only two members
are 2 and negative 2.
00:06:12.920 --> 00:06:15.760
In other words, we cannot
say that 'x squared' is
00:06:15.760 --> 00:06:16.900
a synonym for 4.
00:06:16.900 --> 00:06:22.000
All we can say is given the
condition if 'x' is either 2
00:06:22.000 --> 00:06:25.040
or minus 2, then 'x
squared' equals 4.
00:06:25.040 --> 00:06:27.190
Otherwise, this is
not the case.
00:06:27.190 --> 00:06:31.590
For example, if you subtract 4
from 'x squared', you do not
00:06:31.590 --> 00:06:33.020
get identically 0.
00:06:33.020 --> 00:06:35.610
In other words, 'x squared'
minus 4 is not another
00:06:35.610 --> 00:06:37.210
way of saying 0.
00:06:37.210 --> 00:06:39.320
For example, if I put
'x' equals 3 in
00:06:39.320 --> 00:06:40.650
here, this says what?
00:06:40.650 --> 00:06:43.580
3 squared equals 4, which is
certainly a meaningful
00:06:43.580 --> 00:06:47.250
statement, but nonetheless
a false statement.
00:06:47.250 --> 00:06:50.290
You see, somehow or other, an
equation is something which
00:06:50.290 --> 00:06:55.850
can be true for a certain values
of your variable but
00:06:55.850 --> 00:06:57.160
false for others.
00:06:57.160 --> 00:07:00.110
On the other hand, an identity,
as the name may
00:07:00.110 --> 00:07:03.200
imply, is something that's
true for all
00:07:03.200 --> 00:07:04.690
values of the variable.
00:07:04.690 --> 00:07:06.780
For example, let's go back to
something we learned in
00:07:06.780 --> 00:07:11.850
elementary school algebra: 'x
squared minus 1' equals 'x +
00:07:11.850 --> 00:07:14.150
1' times 'x - 1'.
00:07:14.150 --> 00:07:16.520
Notice I wrote this with
the three lines here.
00:07:16.520 --> 00:07:21.720
It's my way of saying that for
any number 'x' whatsoever, 'x
00:07:21.720 --> 00:07:26.500
squared - 1' always names
the same number as 'x +
00:07:26.500 --> 00:07:28.660
1' times 'x - 1'.
00:07:28.660 --> 00:07:31.860
In other words, the solution
set of this particular
00:07:31.860 --> 00:07:35.860
equation includes all
real numbers.
00:07:35.860 --> 00:07:39.150
Or another way of saying it is
that if you were to subtract
00:07:39.150 --> 00:07:43.940
'x + 1' times 'x - 1' from the
'x squared minus 1', the
00:07:43.940 --> 00:07:46.890
result would be 0 independently
of what the
00:07:46.890 --> 00:07:49.010
value of 'x' was.
00:07:49.010 --> 00:07:52.530
In more colloquial terms, what
we're saying is that 'x
00:07:52.530 --> 00:07:56.830
squared minus 1' and 'x +
1' times 'x - 1' are two
00:07:56.830 --> 00:07:59.510
different ways of saying
the same thing.
00:07:59.510 --> 00:08:03.330
Now, the beauty of an
identity versus a
00:08:03.330 --> 00:08:05.430
conditional equality is this.
00:08:05.430 --> 00:08:08.970
That if two expressions are just
two different names for
00:08:08.970 --> 00:08:12.820
the same thing, then whatever
is true for one expression
00:08:12.820 --> 00:08:15.300
will be true for the other.
00:08:15.300 --> 00:08:17.750
You see, in other words, if the
concept is the same but
00:08:17.750 --> 00:08:20.430
only the names are different,
then certainly anything that
00:08:20.430 --> 00:08:23.050
depends on the concept will not
depend on the particular
00:08:23.050 --> 00:08:23.970
name that's involved.
00:08:23.970 --> 00:08:27.710
Well, as a case in point, let's
suppose now I take 'x
00:08:27.710 --> 00:08:30.270
squared minus 1' and
I differentiate it.
00:08:30.270 --> 00:08:32.580
The result is '2x'.
00:08:32.580 --> 00:08:35.980
On the other hand, if I take
'x + 1' times 'x - 1' and
00:08:35.980 --> 00:08:37.780
differentiate it, I get what?
00:08:37.780 --> 00:08:41.140
By use of the product rule, it's
the first factor times
00:08:41.140 --> 00:08:45.260
the derivative of the second
plus the derivative of the
00:08:45.260 --> 00:08:47.660
first factor times the second.
00:08:47.660 --> 00:08:52.120
That's what? 'x + 1' plus
'x - 1', and that
00:08:52.120 --> 00:08:55.420
comes out to be '2x'.
00:08:55.420 --> 00:08:58.050
In other words, since these two
expressions were just two
00:08:58.050 --> 00:09:01.360
different names for the same
thing, the derivative of one
00:09:01.360 --> 00:09:04.740
of the expressions must equal
the derivative of the other
00:09:04.740 --> 00:09:06.230
because it's still the same
function that you're
00:09:06.230 --> 00:09:07.360
differentiating.
00:09:07.360 --> 00:09:11.130
And this is what we mean when
we say let's assume that 'y'
00:09:11.130 --> 00:09:14.420
is that function of 'x' that
makes the resulting equation
00:09:14.420 --> 00:09:15.640
an identity.
00:09:15.640 --> 00:09:18.450
Let's look at a particularly
simple illustration, and we'll
00:09:18.450 --> 00:09:20.330
do this quite often
in this course.
00:09:20.330 --> 00:09:24.240
Namely, whenever we want to
illustrate a new topic, we
00:09:24.240 --> 00:09:29.270
will always, when possible, pick
an illustrative example
00:09:29.270 --> 00:09:33.170
that could have been solved
by a previous method.
00:09:33.170 --> 00:09:37.040
As a case in point, let's
look at the identity 'x'
00:09:37.040 --> 00:09:40.010
times 'y' is 1.
00:09:40.010 --> 00:09:42.970
See, in other words, 'y' is that
particular function of
00:09:42.970 --> 00:09:48.270
'x', such that whenever you
replace 'y' by that function,
00:09:48.270 --> 00:09:49.365
this becomes an identity.
00:09:49.365 --> 00:09:53.920
Now, if that seems hard, notice
that we could turn this
00:09:53.920 --> 00:09:57.650
into an explicit relationship
just by dividing both sides of
00:09:57.650 --> 00:10:00.310
this equation, our
identity by 'x'.
00:10:00.310 --> 00:10:03.670
Of course, that assumes
'x' is not equal to 0.
00:10:03.670 --> 00:10:06.430
But if we do that, 'y'
becomes '1/x'.
00:10:06.430 --> 00:10:10.600
Now notice that '1/x' has the
property that as long as 'x'
00:10:10.600 --> 00:10:15.490
is not 0, if you multiply that
by 'x', you get identically 1.
00:10:15.490 --> 00:10:17.650
'x' times '1/x' is 1.
00:10:17.650 --> 00:10:20.630
Two different ways
of saying 1.
00:10:20.630 --> 00:10:22.440
Now, how does the method
of implicit
00:10:22.440 --> 00:10:24.060
differentiation proceed?
00:10:24.060 --> 00:10:28.940
We say OK, since this is an
identity, if I differentiate
00:10:28.940 --> 00:10:32.270
both sides with respect to 'x',
the derivative of the
00:10:32.270 --> 00:10:34.910
left-hand side should equal
the derivative of the
00:10:34.910 --> 00:10:36.220
right-hand side.
00:10:36.220 --> 00:10:39.090
Now, how do we differentiate
'x' times 'y'?
00:10:39.090 --> 00:10:42.470
Observe that we're assuming that
'y' is a function of 'x'.
00:10:42.470 --> 00:10:45.490
Therefore, 'x' times 'y'
is a product of two
00:10:45.490 --> 00:10:46.820
functions of 'x'.
00:10:46.820 --> 00:10:49.650
To differentiate a product,
we use the product rule.
00:10:49.650 --> 00:10:53.670
Namely, we will take the first
factor times the derivative of
00:10:53.670 --> 00:10:55.940
the second with respect
to 'x'.
00:10:55.940 --> 00:11:01.030
That's 'dy dx', because 'y' is
the second factor, plus the
00:11:01.030 --> 00:11:03.900
derivative of the first factor
with respect to 'x'.
00:11:03.900 --> 00:11:07.570
Well, the derivative of 'x' with
respect to 'x' is 1 times
00:11:07.570 --> 00:11:10.550
the second factor, which is 'y',
that's now the derivative
00:11:10.550 --> 00:11:11.810
of the left-hand side.
00:11:11.810 --> 00:11:14.870
That must equal identically
the derivative of the
00:11:14.870 --> 00:11:16.120
right-hand side.
00:11:16.120 --> 00:11:17.710
The right-hand side is 1.
00:11:17.710 --> 00:11:19.530
The derivative of
a constant is 0.
00:11:19.530 --> 00:11:23.110
So we wind up with the
relationship that 'x' times
00:11:23.110 --> 00:11:27.190
'dy dx' plus 'y' must
be identically 0.
00:11:27.190 --> 00:11:31.810
And now solving for 'dy dx' in
terms of 'x' and 'y', we find
00:11:31.810 --> 00:11:36.970
that 'dy dx' is equal
to minus 'y/x'.
00:11:36.970 --> 00:11:40.480
By the way, notice that by
picking a problem that could
00:11:40.480 --> 00:11:44.120
be solved explicitly for 'y'
in terms of 'x', we have a
00:11:44.120 --> 00:11:46.920
very simple check on this
particular problem.
00:11:46.920 --> 00:11:50.980
Namely, we know that 'y', if
'x' is not 0, that y is
00:11:50.980 --> 00:11:52.820
another name for '1/x'.
00:11:52.820 --> 00:11:56.540
Therefore, minus 'y/x' is
another name for ''minus 1'
00:11:56.540 --> 00:11:57.900
over 'x squared''.
00:11:57.900 --> 00:12:01.930
But we already know that if 'y'
equals 'x to the minus 1'
00:12:01.930 --> 00:12:06.480
by another method, we know that
'dy dx' is 'minus x to
00:12:06.480 --> 00:12:11.390
the minus 2', which is also
''minus 1' over 'x squared''.
00:12:11.390 --> 00:12:14.820
And so we see that the new
method does give us the same
00:12:14.820 --> 00:12:17.530
answer as the old method.
00:12:17.530 --> 00:12:19.560
By the way, let me
make a rather
00:12:19.560 --> 00:12:22.440
important aside over here.
00:12:22.440 --> 00:12:25.250
And that is when you look at
something like this, you might
00:12:25.250 --> 00:12:28.990
say something like I wonder
what happens if I try to
00:12:28.990 --> 00:12:35.640
compute 'dy dx', say, when 'x'
equals 2 and 'y' equals 3?
00:12:35.640 --> 00:12:37.300
Now, you see if you mechanically
plug into
00:12:37.300 --> 00:12:38.880
something like this,
you get what?
00:12:38.880 --> 00:12:42.710
Minus 3/2, which is minus
three-halves.
00:12:42.710 --> 00:12:45.810
The point that I want to bring
out, and we'll come to this at
00:12:45.810 --> 00:12:49.590
the conclusion of today's
lecture also, is the concept
00:12:49.590 --> 00:12:53.720
of related rates and
related variables.
00:12:53.720 --> 00:12:57.470
Notice that whereas from this
equation it looks as if we can
00:12:57.470 --> 00:13:01.550
let 'y' equal 3 and 'x' equal
2, notice that if we go back
00:13:01.550 --> 00:13:07.000
to our basic definition, 'x'
and 'y' are related so that
00:13:07.000 --> 00:13:08.290
they are not independent.
00:13:08.290 --> 00:13:12.720
Notice that as soon as we say
let 'x' equal 2, we have what?
00:13:12.720 --> 00:13:17.030
That '2y' is 1, and
'y' must be 1/2.
00:13:17.030 --> 00:13:20.740
In other words, when you use
implicit differentiation,
00:13:20.740 --> 00:13:24.150
never forget that whenever
you're going to compare 'x'
00:13:24.150 --> 00:13:29.260
and 'y', you must go back to the
equation or the identity
00:13:29.260 --> 00:13:32.520
which implicitly relates
'x' to 'y'.
00:13:32.520 --> 00:13:35.690
Well, so far this may
look rather easy and
00:13:35.690 --> 00:13:36.760
straightforward.
00:13:36.760 --> 00:13:39.360
But the fact remains that there
are certain subtleties
00:13:39.360 --> 00:13:41.650
here which we have
not hit yet.
00:13:41.650 --> 00:13:45.040
So what I'd like to do now
is pick a second example,
00:13:45.040 --> 00:13:48.230
slightly more complicated than
this one, which can still be
00:13:48.230 --> 00:13:51.950
solved explicitly but which
leads to a wrinkle which we
00:13:51.950 --> 00:13:54.220
may not have observed before.
00:13:54.220 --> 00:13:55.850
With this in mind,
what I would like
00:13:55.850 --> 00:13:58.680
to do is the following.
00:13:58.680 --> 00:14:04.170
Let's consider the relation 'x
squared' plus 'y squared'
00:14:04.170 --> 00:14:09.310
equals 25 and ask the question
how do you find 'dy dx' in
00:14:09.310 --> 00:14:10.830
this particular case?
00:14:10.830 --> 00:14:15.450
Again what we do is we assume,
and this is the big word here.
00:14:15.450 --> 00:14:17.860
There are lots of things that
you can assume, but whether
00:14:17.860 --> 00:14:20.570
they exist or not is
another question.
00:14:20.570 --> 00:14:22.770
That's the part that the
textbook means is more
00:14:22.770 --> 00:14:24.970
advanced and is hard
to justify.
00:14:24.970 --> 00:14:25.920
But let's take a look here.
00:14:25.920 --> 00:14:30.310
We'll assume that 'y' is a
particular function of 'x'
00:14:30.310 --> 00:14:34.380
with the property that when 'y'
is that function of 'x',
00:14:34.380 --> 00:14:39.970
'x squared' plus 'y squared'
is identically 25, OK?
00:14:39.970 --> 00:14:43.040
And what that means is that
whatever 'x' and 'y' are,
00:14:43.040 --> 00:14:45.760
they're related in such a way
that 'x squared' plus 'y
00:14:45.760 --> 00:14:48.780
squared' is a synonym for 25.
00:14:48.780 --> 00:14:52.930
If we now proceed by implicit
differentiation here, you see
00:14:52.930 --> 00:14:57.310
the left-hand side is a function
which is the sum of
00:14:57.310 --> 00:14:59.080
two functions of 'x'.
00:14:59.080 --> 00:15:01.370
The derivative of a sum is the
sum of the derivatives.
00:15:01.370 --> 00:15:04.890
The derivative of 'x squared'
with respect to 'x' is '2x'.
00:15:04.890 --> 00:15:09.440
The derivative of 'y squared'
with respect
00:15:09.440 --> 00:15:11.840
to 'x' is not '2y'.
00:15:11.840 --> 00:15:15.320
The derivative of 'y squared'
with respect to 'y' is '2y'.
00:15:15.320 --> 00:15:18.130
By the chain rule, the
derivative of 'y squared' with
00:15:18.130 --> 00:15:20.080
respect to 'x' is what?
00:15:20.080 --> 00:15:22.480
The derivative of 'y squared'
with respect to
00:15:22.480 --> 00:15:24.760
'y' times 'dy dx'.
00:15:24.760 --> 00:15:26.450
In other words, this
is simply what?
00:15:26.450 --> 00:15:28.860
''2y' 'dy dx''.
00:15:28.860 --> 00:15:31.720
So the derivative of the
left-hand side is '2x' plus
00:15:31.720 --> 00:15:33.160
''2y' 'dy dx''.
00:15:33.160 --> 00:15:35.800
The derivative of the
right-hand side, the
00:15:35.800 --> 00:15:38.020
right-hand side being
a constant, is 0.
00:15:38.020 --> 00:15:41.490
And if we now solve for 'dy dx'
in terms of 'x' and 'y',
00:15:41.490 --> 00:15:47.180
we find that 'dy dx'
is 'minus x/y', OK?
00:15:47.180 --> 00:15:50.210
This is all there is to this
thing mechanically.
00:15:50.210 --> 00:15:53.390
By the way, of course, it
happens as you probably
00:15:53.390 --> 00:15:56.770
remember, that 'x squared' plus
'y squared' equals 25 is
00:15:56.770 --> 00:16:00.850
a circle centered at the origin
with radius equal to 5.
00:16:00.850 --> 00:16:02.520
OK, so far so good.
00:16:02.520 --> 00:16:05.270
We'll come back to this diagram
in a little while.
00:16:05.270 --> 00:16:08.770
But the thing now is could we
have solved the same problem
00:16:08.770 --> 00:16:13.250
by solving for 'y' explicitly
in terms of 'x'?
00:16:13.250 --> 00:16:15.310
The answer in this case,
of course, is yes.
00:16:15.310 --> 00:16:19.220
Namely, if 'x squared' plus 'y
squared' is 25, that says that
00:16:19.220 --> 00:16:22.760
'y squared' is 25 minus 'x
squared', and therefore, the
00:16:22.760 --> 00:16:24.040
desired function of 'x'.
00:16:24.040 --> 00:16:25.740
'y' is what function of 'x'?
00:16:25.740 --> 00:16:28.810
It's plus or minus the
square root of
00:16:28.810 --> 00:16:31.070
'25 minus 'x squared''.
00:16:31.070 --> 00:16:32.210
In other words, that's what?
00:16:32.210 --> 00:16:36.900
It's the positive '25 minus 'x
squared'' to the 1/2 power.
00:16:36.900 --> 00:16:38.640
That's one of the solutions.
00:16:38.640 --> 00:16:44.190
And the other solution is the
negative '25 minus 'x
00:16:44.190 --> 00:16:45.940
squared'' to the 1/2 power.
00:16:45.940 --> 00:16:48.730
I simply use the exponent
notation because it's more
00:16:48.730 --> 00:16:51.560
familiar to us in terms
of differentiation to
00:16:51.560 --> 00:16:54.570
differentiate the exponent
rather than the radical sign.
00:16:54.570 --> 00:16:59.290
But the idea is this: Notice
now that we begin to see a
00:16:59.290 --> 00:17:02.320
multivalued function creeping
in over here.
00:17:02.320 --> 00:17:05.720
In other words, we now find that
we want to solve for 'y'
00:17:05.720 --> 00:17:09.660
explicitly in terms of 'x',
that we do get the problem
00:17:09.660 --> 00:17:12.960
that y might be a multivalued
function of 'x'.
00:17:12.960 --> 00:17:14.900
Well, we'll look at
that in a moment.
00:17:14.900 --> 00:17:17.980
For the time being,
let's simply do a
00:17:17.980 --> 00:17:19.290
double check over here.
00:17:19.290 --> 00:17:22.569
Let's actually differentiate
this thing explicitly and see
00:17:22.569 --> 00:17:24.599
what the derivative
turns out to be.
00:17:24.599 --> 00:17:27.369
If we differentiate this,
we bring the 1/2 down.
00:17:27.369 --> 00:17:30.050
We replace it to an
exponent one less.
00:17:30.050 --> 00:17:35.010
And by the chain rule, we must
multiply by the derivative of
00:17:35.010 --> 00:17:39.200
what's inside here, 25 minus x
squared with respect to 'x'.
00:17:39.200 --> 00:17:40.730
That's 'minus 2x'.
00:17:40.730 --> 00:17:44.630
Collecting terms and
simplifying, we get what?
00:17:44.630 --> 00:17:50.240
'Minus x' over ''25 minus 'x
squared'' to the 1/2'.
00:17:50.240 --> 00:17:54.530
And recalling that ''25 minus
'x squared'' to the 1/2' is
00:17:54.530 --> 00:17:59.070
equal to the 'y' value in this
case, we get the derivative of
00:17:59.070 --> 00:18:04.050
'y1' with respect to 'x' is
'minus x' over 'y1', just
00:18:04.050 --> 00:18:05.260
calling the function this.
00:18:05.260 --> 00:18:07.190
That at any rate shows what?
00:18:07.190 --> 00:18:11.030
The negative x-coordinate over
the y-coordinate, and that
00:18:11.030 --> 00:18:15.840
certainly checks with the result
that we had before.
00:18:15.840 --> 00:18:21.370
In a similar way, if we now
differentiate 'y2' with
00:18:21.370 --> 00:18:25.870
respect to 'x', bringing down
the exponent, multiplying by
00:18:25.870 --> 00:18:28.650
the derivative of what's inside
with respect to 'x', we
00:18:28.650 --> 00:18:31.590
wind up with the same expression
as we did before,
00:18:31.590 --> 00:18:37.040
only now we remember that 'y2'
is defined to be negative ''25
00:18:37.040 --> 00:18:39.920
minus 'x squared'' to the
1/2', so this is really
00:18:39.920 --> 00:18:42.560
negative y2.
00:18:42.560 --> 00:18:46.990
By the way, notice again that we
get the same answer here as
00:18:46.990 --> 00:18:48.770
we did here.
00:18:48.770 --> 00:18:52.650
I like to keep the minus sign
with the appropriate term.
00:18:52.650 --> 00:18:55.540
In other words, notice it was
the fact that 'y2' was
00:18:55.540 --> 00:19:00.570
multivalued that caused us
to have two different
00:19:00.570 --> 00:19:02.700
possibilities here in
the first place.
00:19:02.700 --> 00:19:06.530
Now, let's go back to our little
graph over here and
00:19:06.530 --> 00:19:09.170
take a look to see just
what happened here.
00:19:09.170 --> 00:19:11.930
Remember, all through this
course, we've said what?
00:19:11.930 --> 00:19:15.360
Be aware of what happens
when 'y' equals 0.
00:19:15.360 --> 00:19:19.320
Notice in terms of our picture
and our related rates here,
00:19:19.320 --> 00:19:23.300
when 'y' is 0, 'x' is not
some arbitrary number.
00:19:23.300 --> 00:19:26.430
When 'y' is 0, since 'x squared'
plus 'y squared'
00:19:26.430 --> 00:19:32.200
equals 25 when 'y' is 0, 'x'
must either be 5 or minus 5.
00:19:32.200 --> 00:19:36.880
So what's happening is the bad
point when 'y' is 0, or the
00:19:36.880 --> 00:19:39.290
bad points, are right here.
00:19:39.290 --> 00:19:42.010
And notice the rather
interesting thing here, and
00:19:42.010 --> 00:19:45.480
this is what comes up more
analytically later.
00:19:45.480 --> 00:19:48.920
The only time you're in trouble
when you assume that
00:19:48.920 --> 00:19:51.500
'y' is a function of 'x'--
remember, function mean
00:19:51.500 --> 00:19:52.960
single-valued--
00:19:52.960 --> 00:19:57.690
is that if you happen to be in
the neighborhood of 5 comma 0
00:19:57.690 --> 00:20:01.420
or minus 5 comma 0, notice
that for any neighborhood
00:20:01.420 --> 00:20:04.080
surrounding this point,
there is no way--
00:20:04.080 --> 00:20:07.270
if this point here is included,
there is no way of
00:20:07.270 --> 00:20:10.230
breaking up this function
to be single valued.
00:20:10.230 --> 00:20:12.580
In other words, if you must have
a small portion of the
00:20:12.580 --> 00:20:16.910
curve that includes 5 comma
0 as an interior point, no
00:20:16.910 --> 00:20:20.140
matter how you do it, the
resulting curve is going to be
00:20:20.140 --> 00:20:24.190
multivalued, and then we're in
the same predicament again.
00:20:24.190 --> 00:20:27.080
In other words, from a
theoretical point of view,
00:20:27.080 --> 00:20:31.600
geometrically speaking, it's
very easy to assume that 'y'
00:20:31.600 --> 00:20:34.080
is a function of 'x' that makes
00:20:34.080 --> 00:20:36.620
our equation an identity.
00:20:36.620 --> 00:20:40.370
But from a real point of view,
what may frequently happen is
00:20:40.370 --> 00:20:45.050
that the only type of function
that would work is what we
00:20:45.050 --> 00:20:46.610
call a multivalued function.
00:20:46.610 --> 00:20:48.740
That's exactly what's
going on over here.
00:20:48.740 --> 00:20:51.750
And that's why a textbook,
which is trying to be
00:20:51.750 --> 00:20:55.340
rigorous, has to be very, very
careful in explaining what
00:20:55.340 --> 00:20:58.110
happens in neighborhoods
of points like this.
00:20:58.110 --> 00:21:01.205
But again, we'll talk about
that in more detail in a
00:21:01.205 --> 00:21:02.130
little while.
00:21:02.130 --> 00:21:04.880
I would like to make one
aside over here.
00:21:04.880 --> 00:21:09.710
You may recall that we learned
in one of our previous
00:21:09.710 --> 00:21:12.880
lectures that by inverse
functions, we can
00:21:12.880 --> 00:21:16.840
differentiate things like 'x'
to the 1/2, 'x' to the 1/3.
00:21:16.840 --> 00:21:20.830
This generalizes very nicely
and very simply in terms of
00:21:20.830 --> 00:21:22.680
implicit differentiation.
00:21:22.680 --> 00:21:27.310
Namely, suppose you have 'y'
equals 'x to the 'p/q'' power
00:21:27.310 --> 00:21:29.340
where 'p' and 'q'
are integers.
00:21:29.340 --> 00:21:33.160
In other words, your exponent
is now a fraction.
00:21:33.160 --> 00:21:38.160
Raise both sides to the q-th
power, and you get 'y' to the
00:21:38.160 --> 00:21:40.560
'q' equals 'x to the p'.
00:21:40.560 --> 00:21:44.450
Now, we know how to
differentiate powers of 'x' or
00:21:44.450 --> 00:21:47.720
'y' if the power happens
to be an integer.
00:21:47.720 --> 00:21:50.950
So by implicit differentiation,
assuming that
00:21:50.950 --> 00:21:53.960
this is now an identity, which
it is if 'y' equals 'x to the
00:21:53.960 --> 00:21:57.370
'p/q'', if we differentiate
both sides with respect to
00:21:57.370 --> 00:22:01.890
'x', the derivative of the
left-hand side is 'qy to the
00:22:01.890 --> 00:22:05.670
'q - 1'' times the derivative
of 'y' with respect to 'x'.
00:22:05.670 --> 00:22:08.110
That is, we're differentiating
with respect to 'x'.
00:22:08.110 --> 00:22:10.180
And the derivative of the
right-hand side is
00:22:10.180 --> 00:22:12.790
'px to the 'p - 1''.
00:22:12.790 --> 00:22:15.710
And since this is an
identity, these two
00:22:15.710 --> 00:22:17.360
expressions must be equal.
00:22:17.360 --> 00:22:21.940
If we solve now for 'dy dx',
we obtain this expression,
00:22:21.940 --> 00:22:27.780
replacing 'y' by 'x to the
'p/q'', and multiplying out
00:22:27.780 --> 00:22:31.980
here, and remembering that when
you have an exponent in
00:22:31.980 --> 00:22:34.700
the denominator, it can come
up into the numerator by
00:22:34.700 --> 00:22:36.540
changing the sign.
00:22:36.540 --> 00:22:41.060
A little bit of arithmetic shows
that 'dy dx' is ''p/q' x
00:22:41.060 --> 00:22:47.780
to the ''p - 1' minus 'p'
plus 'p/q'' power.
00:22:47.780 --> 00:22:51.180
The 'p' and the 'minus p' cancel
out, and we find what?
00:22:51.180 --> 00:22:57.090
That 'dy dx' is 'p/q' times
'x to the ''p/q' - 1''.
00:22:57.090 --> 00:23:00.760
Remembering that 'p/q' was
our fractional exponent,
00:23:00.760 --> 00:23:02.350
we again see what?
00:23:02.350 --> 00:23:05.150
That to differentiate 'x' to
a fractional exponent--
00:23:05.150 --> 00:23:06.260
'p/q'--
00:23:06.260 --> 00:23:10.250
you bring the exponent down and
replace the 2 by one less.
00:23:10.250 --> 00:23:13.380
So that gives us an alternative
method for finding
00:23:13.380 --> 00:23:17.680
the derivative of a fractional
exponent by namely using
00:23:17.680 --> 00:23:19.580
implicit differentiation.
00:23:19.580 --> 00:23:22.040
Now, the reason I put that aside
in is we could certainly
00:23:22.040 --> 00:23:23.180
get along without it.
00:23:23.180 --> 00:23:27.170
But I felt that before we go any
further, that I would like
00:23:27.170 --> 00:23:30.580
at least to add it on the more
elementary levels to be sure
00:23:30.580 --> 00:23:34.110
that we understand explicitly
what implicit
00:23:34.110 --> 00:23:36.720
differentiation means.
00:23:36.720 --> 00:23:41.000
With this in mind, let's now go
back to the problem that we
00:23:41.000 --> 00:23:45.890
started off with in this
lecture, namely, the curve
00:23:45.890 --> 00:23:48.760
whose equation is 'x to the
eighth' plus ''x to the sixth'
00:23:48.760 --> 00:23:52.390
'y to the fourth'' plus 'y
the sixth' equals 3.
00:23:52.390 --> 00:23:53.340
And we say what?
00:23:53.340 --> 00:23:56.630
Let's find the equation of the
line tangent to this curve at
00:23:56.630 --> 00:23:58.470
the point 1 comma 1.
00:23:58.470 --> 00:24:00.790
By the way, I did something
that teachers are
00:24:00.790 --> 00:24:01.950
allowed to do here.
00:24:01.950 --> 00:24:04.620
I rigged the problem to come
out rather simply.
00:24:04.620 --> 00:24:08.250
You see, what I did was by
picking 1 and 1 here,
00:24:08.250 --> 00:24:09.200
it turns out what?
00:24:09.200 --> 00:24:12.280
That if I made the right-hand
side just equal to the number
00:24:12.280 --> 00:24:15.730
of terms here, I would get
this thing to check out.
00:24:15.730 --> 00:24:18.160
But notice, by the way, if I had
just picked the point at
00:24:18.160 --> 00:24:22.700
random to try to even check
where that point is on this
00:24:22.700 --> 00:24:27.720
curve, or if it were on this
curve, to find out what its
00:24:27.720 --> 00:24:31.500
coordinates are, this is a
rather difficult problem.
00:24:31.500 --> 00:24:34.600
In other words, if I just pick
a random value of 'x', as I
00:24:34.600 --> 00:24:37.560
said before, I get a sixth
degree polynomial equation to
00:24:37.560 --> 00:24:42.260
solve, and if I pick a random
value of 'y', an eighth degree
00:24:42.260 --> 00:24:43.770
polynomial equation to solve.
00:24:43.770 --> 00:24:46.480
You know, it might be nice to
pretend that the 8 was a 2 or
00:24:46.480 --> 00:24:49.100
something like this and solve
it as a quadratic equation.
00:24:49.100 --> 00:24:52.770
But quite frankly, solving
polynomial equations, which
00:24:52.770 --> 00:24:56.940
are higher than the degree two
is a very difficult task.
00:24:56.940 --> 00:25:00.500
In fact, if the degree is
greater than four, it may even
00:25:00.500 --> 00:25:02.510
be an impossible task.
00:25:02.510 --> 00:25:04.140
I'm not going to go into
that now because
00:25:04.140 --> 00:25:05.280
it's not that crucial.
00:25:05.280 --> 00:25:09.170
What is crucial is to observe
that this particular problem
00:25:09.170 --> 00:25:10.870
makes sense.
00:25:10.870 --> 00:25:12.890
It's a meaningful problem.
00:25:12.890 --> 00:25:16.600
It would be difficult, if not
impossible, to solve for 'y'
00:25:16.600 --> 00:25:19.690
explicitly in terms
of 'x' here, OK?
00:25:19.690 --> 00:25:23.430
Yet to solve this problem,
we can now use implicit
00:25:23.430 --> 00:25:27.030
differentiation except for the
fact that we can no longer
00:25:27.030 --> 00:25:29.950
check back by another method to
see if the answer is right.
00:25:29.950 --> 00:25:33.150
That's why I wanted to pick a
few introductory examples that
00:25:33.150 --> 00:25:35.240
we could check by other means.
00:25:35.240 --> 00:25:37.800
Now that we have a feeling for
this, maybe we will trust the
00:25:37.800 --> 00:25:40.880
technique in a case where we
have no recourse other than to
00:25:40.880 --> 00:25:42.660
use the technique.
00:25:42.660 --> 00:25:46.660
So what we do now is this: We
say OK, let's assume that 'y'
00:25:46.660 --> 00:25:50.540
is that function of 'x', that
differentiable function of 'x'
00:25:50.540 --> 00:25:54.760
that makes this particular
equation an identity.
00:25:54.760 --> 00:25:57.780
And assuming now that this
is an identity, let me
00:25:57.780 --> 00:26:01.810
differentiate both sides
with respect to 'x'.
00:26:01.810 --> 00:26:03.410
See, this is a sum.
00:26:03.410 --> 00:26:05.850
One of the terms in the sum
happens to be a product.
00:26:05.850 --> 00:26:09.220
But by now, hopefully how we go
about something like this
00:26:09.220 --> 00:26:10.330
will be old hat.
00:26:10.330 --> 00:26:13.180
Namely, to differentiate the
left-hand side with respect to
00:26:13.180 --> 00:26:14.710
'x', we get what?
00:26:14.710 --> 00:26:18.750
'8x to the seventh' plus what?
00:26:18.750 --> 00:26:21.580
The derivative of 'x to the
sixth', which is '6x to the
00:26:21.580 --> 00:26:24.230
fifth', times the second
factor, which is 'y the
00:26:24.230 --> 00:26:27.050
fourth', plus the first factor,
which is 'x to the
00:26:27.050 --> 00:26:30.090
sixth', times the derivative
of 'y to the fourth' with
00:26:30.090 --> 00:26:31.150
respect to 'x'--
00:26:31.150 --> 00:26:33.800
that's ''4y cubed' 'dy dx''--
00:26:33.800 --> 00:26:36.060
plus the derivative of 'y to
the sixth' with respect to
00:26:36.060 --> 00:26:39.700
'x', which is ''6y to
the fifth' 'dy dx''.
00:26:39.700 --> 00:26:42.210
That must be identically 0.
00:26:42.210 --> 00:26:45.710
And if we now solve this for 'dy
dx', just by transposing,
00:26:45.710 --> 00:26:50.320
we wind up again with a rather
messy expression, but which
00:26:50.320 --> 00:26:55.770
does show what 'dy dx' looks
like in terms of 'x' and 'y'.
00:26:55.770 --> 00:26:59.830
We were interested in knowing
what the slope was not only
00:26:59.830 --> 00:27:03.900
for any old value of 'x' and
'y' but rather for what?
00:27:03.900 --> 00:27:06.860
When 'x' is 1 and 'y' is 1.
00:27:06.860 --> 00:27:09.280
And that works out very nicely
computationally.
00:27:09.280 --> 00:27:11.490
It's just minus 14/10.
00:27:11.490 --> 00:27:14.960
In other words, at the point
1 comma 1, the slope of the
00:27:14.960 --> 00:27:17.370
curve is minus 7/5.
00:27:17.370 --> 00:27:20.240
The curve passes through
the point (1, 1).
00:27:20.240 --> 00:27:24.460
Hence, it's equation is 'y - 1'
over 'x - 1' equals minus
00:27:24.460 --> 00:27:30.660
7/5, or more explicitly,
'7x + 5y' equals 12.
00:27:30.660 --> 00:27:33.660
And the point that I wanted to
make here is notice that
00:27:33.660 --> 00:27:38.090
nothing changed in principle
from our first few lectures.
00:27:38.090 --> 00:27:41.220
Notice that to find the equation
of the line, we still
00:27:41.220 --> 00:27:44.090
use the recipe that we have
to know a point on the
00:27:44.090 --> 00:27:45.760
line and the slope.
00:27:45.760 --> 00:27:48.920
The only thing that's changed
with today's lesson is that we
00:27:48.920 --> 00:27:53.140
can now find the slope of a
curve at a particular point
00:27:53.140 --> 00:27:56.160
that we could not find prior
to today's lesson.
00:27:56.160 --> 00:27:59.560
All that has changed is that we
have one more technique for
00:27:59.560 --> 00:28:03.950
finding the derivative of a
particular type of function.
00:28:03.950 --> 00:28:06.710
I would like to analyze this
problem in more detail.
00:28:06.710 --> 00:28:10.680
In particular, I would like to
see where the numerator of
00:28:10.680 --> 00:28:13.780
this expression can
be 0 and where the
00:28:13.780 --> 00:28:15.330
denominator can be 0.
00:28:15.330 --> 00:28:18.390
Because you see in terms of
slopes, where the numerator is
00:28:18.390 --> 00:28:21.770
0, it means the slope
will be 0.
00:28:21.770 --> 00:28:24.110
That means we have a horizontal
tangent line.
00:28:24.110 --> 00:28:27.800
Where the denominator is 0,
that means the slope is
00:28:27.800 --> 00:28:29.890
infinite, OK?
00:28:29.890 --> 00:28:31.820
And where the slope is infinite,
that means you have
00:28:31.820 --> 00:28:34.770
a vertical line, and that means
you have a vertical
00:28:34.770 --> 00:28:36.490
tangent, OK.
00:28:36.490 --> 00:28:38.450
So let's take a look and
see what that means.
00:28:38.450 --> 00:28:42.420
Keep this particular equation
in mind, because now you see
00:28:42.420 --> 00:28:45.900
on the next board, all I want
to do is work with what this
00:28:45.900 --> 00:28:46.420
thing means.
00:28:46.420 --> 00:28:51.780
In other words, 'dy dx' will be
0 when my numerator is 0.
00:28:51.780 --> 00:28:54.810
My numerator can be written
in this particular form.
00:28:54.810 --> 00:28:57.420
And by the way, here's again
an interesting point.
00:28:57.420 --> 00:28:59.800
When will this expression
be 0?
00:28:59.800 --> 00:29:03.810
And the answer is when either
of these two factors is 0.
00:29:03.810 --> 00:29:07.260
Well, the first factor
is 0 when 'x' is 0.
00:29:07.260 --> 00:29:11.630
And the second factor is 0,
since these are even powers,
00:29:11.630 --> 00:29:14.460
only when 'x' and
'y' are both 0.
00:29:14.460 --> 00:29:18.870
Notice, however, that 'x' and
'y' cannot both be 0.
00:29:18.870 --> 00:29:22.420
Recall that the equation
was what?
00:29:22.420 --> 00:29:25.900
Whatever it was, notice that
(0, 0) is not a point which
00:29:25.900 --> 00:29:28.110
satisfies the equation.
00:29:28.110 --> 00:29:32.980
Remember, (0, 0) does not
satisfy 'x to the eighth' plus
00:29:32.980 --> 00:29:35.760
''x to the sixth' 'y squared''
plus 'y to the
00:29:35.760 --> 00:29:37.510
sixth' equals 3.
00:29:37.510 --> 00:29:38.830
It's equal to 0, you see.
00:29:38.830 --> 00:29:42.030
But at any rate, notice
that the slope is 0
00:29:42.030 --> 00:29:43.990
only when 'x' is 0.
00:29:43.990 --> 00:29:47.970
And because of the particular
equation, when 'x' is 0--
00:29:47.970 --> 00:29:50.100
well, we'll go back to
that in a minute.
00:29:50.100 --> 00:29:52.410
Let's just check to see
what's happening here.
00:29:52.410 --> 00:29:53.760
The denominator will be 0.
00:29:53.760 --> 00:29:56.890
In other words, when 'dx
dy' is 0-- '1 over
00:29:56.890 --> 00:29:58.650
'dy dx'', you see--
00:29:58.650 --> 00:30:02.265
only when this factor here is 0,
and that occurs again only
00:30:02.265 --> 00:30:03.697
when 'y' is 0.
00:30:03.697 --> 00:30:06.440
Now, the point to keep
in mind is this.
00:30:06.440 --> 00:30:09.460
Remember, I put this down here
so we could refer to it, and I
00:30:09.460 --> 00:30:11.690
forgot that I put it here, and
that's why I didn't see it
00:30:11.690 --> 00:30:12.810
until just now.
00:30:12.810 --> 00:30:17.400
All I'm saying here is that
notice that when 'x' is 0, 'y'
00:30:17.400 --> 00:30:20.880
must be plus or minus
the sixth root of 3.
00:30:20.880 --> 00:30:24.570
And when 'y' is 0, these two
terms drop out. 'x' must be
00:30:24.570 --> 00:30:27.740
plus or minus the eighth
root of 3.
00:30:27.740 --> 00:30:35.190
Coming over to a graph here
then, what we see is that the
00:30:35.190 --> 00:30:40.260
curve crosses the x-axis
at this point
00:30:40.260 --> 00:30:41.880
with a vertical tangent.
00:30:41.880 --> 00:30:44.660
It crosses the y-axis
at this point with
00:30:44.660 --> 00:30:47.910
a horizontal tangent.
00:30:47.910 --> 00:30:51.650
Notice, by the way, that this
curve is also symmetric with
00:30:51.650 --> 00:30:55.470
respect to both the x- and
the y-axes, because
00:30:55.470 --> 00:30:57.330
if I replace 'x'--
00:30:57.330 --> 00:31:00.990
well, it's not important, and
I don't want to obscure the
00:31:00.990 --> 00:31:05.870
lecture by taking time
out for this now.
00:31:05.870 --> 00:31:09.100
But the point is a quick check
shows that this curve is
00:31:09.100 --> 00:31:12.990
symmetric with respect to both
the x- and the y-axes, that if
00:31:12.990 --> 00:31:15.650
I could plot this curve just
in the first quadrant, the
00:31:15.650 --> 00:31:18.730
mirror image with respect to the
y-axis would then show me
00:31:18.730 --> 00:31:19.970
the second quadrant.
00:31:19.970 --> 00:31:24.170
If I then took the mirror image
of this upper half with
00:31:24.170 --> 00:31:27.180
respect to the x-axis, that
would give me the lower
00:31:27.180 --> 00:31:28.440
portion of this curve.
00:31:28.440 --> 00:31:31.230
The curve tends to look
something like this.
00:31:31.230 --> 00:31:34.530
And by the way, all we've done,
if we look back over
00:31:34.530 --> 00:31:39.330
here if you can see this OK, the
'7x + 5y' equals 12 simply
00:31:39.330 --> 00:31:40.630
turned out to be what?
00:31:40.630 --> 00:31:45.270
The equation of the line which
was tangent to this curve at
00:31:45.270 --> 00:31:46.870
this particular point.
00:31:46.870 --> 00:31:49.850
What I'd like to show you in
terms of one-to-oneness and
00:31:49.850 --> 00:31:53.530
single-valuedness is this.
00:31:53.530 --> 00:31:56.130
I told you I was interested in
what was happening at this
00:31:56.130 --> 00:31:58.750
curve at the point 1 comma 1.
00:31:58.750 --> 00:32:02.690
Suppose I had said instead find
the equation of the line
00:32:02.690 --> 00:32:07.130
tangent to this curve at the
point whose x-coordinate is 1?
00:32:07.130 --> 00:32:09.850
Well, you see, there are two
points on this curve whose
00:32:09.850 --> 00:32:12.700
x-coordinate is 1.
00:32:12.700 --> 00:32:16.170
You see, this is a double-valued
curve.
00:32:16.170 --> 00:32:19.940
A given value of 'x' between
these two extremes yields the
00:32:19.940 --> 00:32:21.300
two values of 'y'.
00:32:21.300 --> 00:32:23.910
I would have had no way of
knowing which of the two
00:32:23.910 --> 00:32:26.580
y-values I meant, you see?
00:32:26.580 --> 00:32:30.090
And correspondingly, if somebody
had said find the
00:32:30.090 --> 00:32:33.290
slope of the curve, of the
equation of a line tangent to
00:32:33.290 --> 00:32:37.920
the curve, at the point whose
y-coordinate is 1, notice that
00:32:37.920 --> 00:32:39.950
this is not 1 to 1.
00:32:39.950 --> 00:32:43.430
In other words, if the
y-coordinate is 1, notice that
00:32:43.430 --> 00:32:47.980
I cannot distinguish between
the point 1 comma 1 and the
00:32:47.980 --> 00:32:49.240
point what?
00:32:49.240 --> 00:32:51.810
Minus 1 comma 1.
00:32:51.810 --> 00:32:55.120
You see, there's again our
problem with inverse functions
00:32:55.120 --> 00:32:56.770
and things of this
particular type.
00:32:56.770 --> 00:32:59.830
If we're told the neighborhood
of the point that we're
00:32:59.830 --> 00:33:01.760
interested in, we're fine.
00:33:01.760 --> 00:33:05.180
If all we're told are one of
the coordinates and have to
00:33:05.180 --> 00:33:08.180
find the other, there is a
certain amount of ambiguity.
00:33:08.180 --> 00:33:11.620
And keep in mind, by the way,
that I deliberately rigged
00:33:11.620 --> 00:33:14.610
this problem to get something
I could graph at least.
00:33:14.610 --> 00:33:18.010
In many cases, it's much more
difficult to even visualize
00:33:18.010 --> 00:33:21.080
what the graph looks like,
much more complicated.
00:33:21.080 --> 00:33:24.060
You see, computationally, this
can be come quite a mess.
00:33:24.060 --> 00:33:27.370
The important point is that what
we were doing implicitly
00:33:27.370 --> 00:33:33.280
here assumed on the explicit
fact I could take this curve
00:33:33.280 --> 00:33:36.270
and break it down at the points
where I have vertical
00:33:36.270 --> 00:33:40.380
tangents and look at this as
two separate curves: 'c1',
00:33:40.380 --> 00:33:44.390
namely, the original curve, but
restricted to 'y' being
00:33:44.390 --> 00:33:48.610
non-negative, and 'c2', the
original curve, but restricted
00:33:48.610 --> 00:33:49.990
to 'y' being negative.
00:33:49.990 --> 00:33:54.880
In other words, I can look at
'c1' as being this piece and
00:33:54.880 --> 00:33:58.360
'c2' as being this piece.
00:33:58.360 --> 00:34:00.550
And again, the point is what?
00:34:00.550 --> 00:34:03.840
That whenever you're in the
neighborhood of these points,
00:34:03.840 --> 00:34:04.840
you're in trouble.
00:34:04.840 --> 00:34:07.340
Because notice that no matter
how small a neighborhood I
00:34:07.340 --> 00:34:12.219
pick, if I can't tell one of
these branches from the other,
00:34:12.219 --> 00:34:15.170
no matter how I do this, I'm
going to be caught on a
00:34:15.170 --> 00:34:18.409
multivalued part of the
curve over here.
00:34:18.409 --> 00:34:22.310
Well, at any rate, I think this
begins to show us what
00:34:22.310 --> 00:34:25.199
implicit differentiation means,
why we have to be
00:34:25.199 --> 00:34:28.780
careful of points at which
vertical tangent occur, but I
00:34:28.780 --> 00:34:31.949
would like before closing this
lecture to generalize the
00:34:31.949 --> 00:34:35.650
concept of related rates and
make this a little bit more
00:34:35.650 --> 00:34:38.719
applicable from a physical
point of view.
00:34:38.719 --> 00:34:41.949
And that is when we say in
something like this that let's
00:34:41.949 --> 00:34:44.449
assume that 'y' is a
differentiable function of
00:34:44.449 --> 00:34:48.380
'x', there is no reason to
have to assume that the
00:34:48.380 --> 00:34:51.929
variable you want to relate
things to is 'x' itself.
00:34:51.929 --> 00:34:56.620
For example, let me do something
which uses the same
00:34:56.620 --> 00:34:59.370
kind of an equation that we had
before, but only from a
00:34:59.370 --> 00:35:00.760
different point of view.
00:35:00.760 --> 00:35:03.820
See, now instead of asking for
the slope of this circle or
00:35:03.820 --> 00:35:06.560
what have you, let's work
the question this way.
00:35:06.560 --> 00:35:08.450
Let's suppose we have
a particle.
00:35:08.450 --> 00:35:11.130
The particle is moving along the
curve 'x squared' plus 'y
00:35:11.130 --> 00:35:14.620
squared' equals 25 where for
physical reasons we'll say 'x'
00:35:14.620 --> 00:35:18.330
and 'y' are in feet, and that
we know at the point 3 comma
00:35:18.330 --> 00:35:20.940
4, 'dx dt'--
00:35:20.940 --> 00:35:24.110
the horizontal component of the
speed of the particle--
00:35:24.110 --> 00:35:25.670
is 8 feet per second.
00:35:25.670 --> 00:35:28.550
And the question is we'd
like to find 'dy dt',
00:35:28.550 --> 00:35:29.455
the vertical component.
00:35:29.455 --> 00:35:32.810
In other words, the particle
is moving along the circle.
00:35:32.810 --> 00:35:37.620
We know that at the point 3
comma 4, 'x' is increasing at
00:35:37.620 --> 00:35:40.630
the rate of 6 feet per second
while the increasing
00:35:40.630 --> 00:35:42.180
x-direction is this way.
00:35:42.180 --> 00:35:44.190
So somehow or other, we know
that the particle is moving
00:35:44.190 --> 00:35:46.620
along the curve in
this direction.
00:35:46.620 --> 00:35:49.340
You see, if it were moving
in this direction, it's
00:35:49.340 --> 00:35:53.280
x-coordinate would be
decreasing, not increasing.
00:35:53.280 --> 00:35:56.040
And the question that comes
up is how do we find
00:35:56.040 --> 00:35:57.860
'dy dt' in this case?
00:35:57.860 --> 00:36:00.480
Well, see, all we do in this
case is we say look-it,
00:36:00.480 --> 00:36:03.770
instead of assuming that 'y' is
a differentiable function
00:36:03.770 --> 00:36:07.520
of 'x', why don't we assume
that both 'x' and 'y' are
00:36:07.520 --> 00:36:09.860
differentiable functions
of 't'?
00:36:09.860 --> 00:36:12.230
In other words, let's assume
that 'x' and 'y' are
00:36:12.230 --> 00:36:21.100
differentiable functions of 't'
that make this an identity
00:36:21.100 --> 00:36:22.540
in terms of 't'.
00:36:22.540 --> 00:36:26.190
Now again, as long as this is an
identity and we're assuming
00:36:26.190 --> 00:36:28.070
that 'x' and 'y' are
differentiable functions of
00:36:28.070 --> 00:36:30.990
't', there is absolutely
no reason why we can't
00:36:30.990 --> 00:36:34.790
differentiate both sides of this
equation with respect to
00:36:34.790 --> 00:36:37.120
't' instead of with
respect to 'x'.
00:36:37.120 --> 00:36:38.940
If we do that, we get what?
00:36:38.940 --> 00:36:44.630
'2x' 'dx dt' plus '2y'
'dy dt' equals 0.
00:36:44.630 --> 00:36:47.610
You see the same principle as
before even though we're now
00:36:47.610 --> 00:36:50.830
differentiating implicitly
evidently what we could call
00:36:50.830 --> 00:36:52.160
parametric equations.
00:36:52.160 --> 00:36:54.880
We're assuming that 'x' and
'y' are differentiable
00:36:54.880 --> 00:36:56.700
functions of 't'.
00:36:56.700 --> 00:37:00.320
You see, from this identity now,
we can conclude that 'dy
00:37:00.320 --> 00:37:04.950
dt' is 'minus x/y'
times 'dx dt'.
00:37:04.950 --> 00:37:09.610
And now, you see, to polish off
our particular problem,
00:37:09.610 --> 00:37:11.300
namely, we're interested
in what?
00:37:11.300 --> 00:37:17.240
When 'x' was 3, 'y' was
4, and 'dx dt' is 8.
00:37:17.240 --> 00:37:21.190
In other words, I find out
from this that 'dy dt' is
00:37:21.190 --> 00:37:24.310
minus 6 feet per second.
00:37:24.310 --> 00:37:29.240
You see, the idea was that 'x'
and 'y' as positions of the
00:37:29.240 --> 00:37:33.070
point were related by the fact
that 'x squared' plus 'y
00:37:33.070 --> 00:37:35.710
squared' had to equal 25.
00:37:35.710 --> 00:37:37.960
In other words, this is what
we physically call a
00:37:37.960 --> 00:37:39.020
constraint.
00:37:39.020 --> 00:37:41.860
And by the way, this is what
makes calculus such an
00:37:41.860 --> 00:37:44.370
important, powerful,
analytical tool.
00:37:44.370 --> 00:37:47.320
Notice that in doing this
particular problem, I never
00:37:47.320 --> 00:37:50.500
had to know explicitly
what functions 'x'
00:37:50.500 --> 00:37:52.000
and 'y' were of 't'.
00:37:52.000 --> 00:37:53.600
All I had to know was what?
00:37:53.600 --> 00:37:56.140
That 'x' and 'y' were
differentiable
00:37:56.140 --> 00:37:57.770
functions of 't'.
00:37:57.770 --> 00:38:01.030
I don't care how the particle
was moving away from the point
00:38:01.030 --> 00:38:04.160
3 comma 4 as long as 'x' and
'y' were differentiable
00:38:04.160 --> 00:38:09.130
functions of 't', this is how
the rates 'dy dt' and 'dx dt'
00:38:09.130 --> 00:38:10.300
had to be related.
00:38:10.300 --> 00:38:13.220
Well, let me summarize what
was really important
00:38:13.220 --> 00:38:15.890
conceptually about
today's lecture.
00:38:15.890 --> 00:38:18.340
The most important thing
conceptually was what?
00:38:18.340 --> 00:38:22.110
With all of our talk about
explicitly writing an output
00:38:22.110 --> 00:38:25.400
and an input, in many important
mathematical
00:38:25.400 --> 00:38:29.810
relationships, the variables
that we're concerned with are
00:38:29.810 --> 00:38:31.670
implicitly related.
00:38:31.670 --> 00:38:34.170
In other words, we are not told
what 'y' looks like in
00:38:34.170 --> 00:38:37.880
terms of 'x', but rather how 'x'
and 'y' are interrelated,
00:38:37.880 --> 00:38:40.680
and from here we have to
find a derivative.
00:38:40.680 --> 00:38:44.190
And the way we do this is that
we make the assumption that an
00:38:44.190 --> 00:38:47.910
appropriate identity exists,
that 'y' is an appropriate
00:38:47.910 --> 00:38:51.310
function of 'x' that makes the
relationship an identity.
00:38:51.310 --> 00:38:54.350
The validity of when you can
do this, for example,
00:38:54.350 --> 00:38:57.660
geometrically, when do you wind
up not having a point
00:38:57.660 --> 00:39:00.730
that includes a vertical
tangent, a point where you
00:39:00.730 --> 00:39:02.910
can't separate the curve
doubling back?
00:39:02.910 --> 00:39:05.780
That turns out to be a rather
difficult point from an
00:39:05.780 --> 00:39:08.465
analytical point of view, a
point that we will return to
00:39:08.465 --> 00:39:11.860
in a more advanced context when
we deal with functions of
00:39:11.860 --> 00:39:13.030
several variables.
00:39:13.030 --> 00:39:15.850
But for the time being, all I
want you to be left with in
00:39:15.850 --> 00:39:20.020
this lecture is the feeling that
we can, given an implicit
00:39:20.020 --> 00:39:23.770
relationship under the proper
conditions, assume that the
00:39:23.770 --> 00:39:26.870
appropriate explicit
relationship exists and that
00:39:26.870 --> 00:39:30.540
we can differentiate both
sides as an identity.
00:39:30.540 --> 00:39:33.700
Well, enough said for today, and
until next time, goodbye.
00:39:36.740 --> 00:39:39.270
NARRATOR: Funding for the
publication of this video was
00:39:39.270 --> 00:39:43.990
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:39:43.990 --> 00:39:48.170
Help OCW continue to provide
free and open access to MIT
00:39:48.170 --> 00:39:52.360
courses by making a donation
at ocw.mit.edu/donate.