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PROFESSOR: Hi.
00:00:33.430 --> 00:00:36.300
Welcome once again to our
lectures in Calculus
00:00:36.300 --> 00:00:39.440
Revisited, where today we are
going to talk about the
00:00:39.440 --> 00:00:42.350
calculus of composite
functions.
00:00:42.350 --> 00:00:46.120
Now, recall that we have already
mentioned in previous
00:00:46.120 --> 00:00:49.420
lectures the notion of
a composite function.
00:00:49.420 --> 00:00:53.490
And what we're going to do today
is to emphasize the idea
00:00:53.490 --> 00:00:58.250
as to how often we are called
upon to find functional
00:00:58.250 --> 00:01:01.500
relationships, where the first
variable is given in terms of
00:01:01.500 --> 00:01:04.730
a second variable, and the
second variable, say, is given
00:01:04.730 --> 00:01:06.590
in terms of the third
variable.
00:01:06.590 --> 00:01:09.210
And we wish to find, say,
the first variable in
00:01:09.210 --> 00:01:10.630
terms of the third.
00:01:10.630 --> 00:01:13.950
Fact here is where the name "the
chain rule" seems to come
00:01:13.950 --> 00:01:17.770
from, a chain reaction where the
variables are related in a
00:01:17.770 --> 00:01:18.940
chain this way.
00:01:18.940 --> 00:01:22.570
Now, we can see this quite
easily in terms of a diagram.
00:01:22.570 --> 00:01:25.930
Suppose, for example,
that I have a graph
00:01:25.930 --> 00:01:28.330
of 'x' versus 'y'.
00:01:28.330 --> 00:01:33.970
And suppose also, I have a
graph of 't' versus 'x'.
00:01:33.970 --> 00:01:35.850
Without any reference
to calculus--
00:01:35.850 --> 00:01:37.210
and this is rather important--
00:01:37.210 --> 00:01:40.470
without any reference to
calculus, notice that these
00:01:40.470 --> 00:01:45.010
two graphs together allow
me to visualize
00:01:45.010 --> 00:01:47.330
'y' in terms of 't'.
00:01:47.330 --> 00:01:50.430
For example, given a particular
value of 't'--
00:01:50.430 --> 00:01:52.210
let's call it 't sub 1'--
00:01:52.210 --> 00:01:57.570
given a value of 't', from that
value of 't', I can find
00:01:57.570 --> 00:01:59.710
the corresponding
value of 'x'.
00:01:59.710 --> 00:02:01.780
Let's call that 'x sub 1'.
00:02:01.780 --> 00:02:06.330
Now, knowing 'x sub 1', I can
come to this diagram.
00:02:06.330 --> 00:02:10.539
Knowing what 'x sub 1' is,
I can find 'y sub 1'.
00:02:10.539 --> 00:02:14.590
And so you see in this chain of
two diagrams, a particular
00:02:14.590 --> 00:02:20.670
value of 't' allows me to find
a particular value of 'y'.
00:02:20.670 --> 00:02:25.560
And in this particular way,
I can visualize 'y' as a
00:02:25.560 --> 00:02:27.420
function of 't'.
00:02:27.420 --> 00:02:30.550
And you see at this stage of the
game, there is absolutely
00:02:30.550 --> 00:02:34.200
no need to have to have any
knowledge of calculus to
00:02:34.200 --> 00:02:37.960
understand what it is that
we're discussing.
00:02:37.960 --> 00:02:41.840
The place that calculus comes
in is in the following way.
00:02:41.840 --> 00:02:45.400
Let's suppose it happened in
this first diagram that the
00:02:45.400 --> 00:02:48.290
graph of 'y' versus
'x' was smooth.
00:02:48.290 --> 00:02:51.430
In other words, let's assume
that 'y' is a differentiable
00:02:51.430 --> 00:02:52.580
function of 'x'.
00:02:52.580 --> 00:02:55.690
In particular, the way I've
drawn this diagram here, we're
00:02:55.690 --> 00:03:01.810
saying suppose 'dy dx' evaluated
at 'x' equals 'x1'
00:03:01.810 --> 00:03:03.850
happens to exist.
00:03:03.850 --> 00:03:09.220
And suppose also that this graph
of 'x' versus 't', this
00:03:09.220 --> 00:03:12.460
also happens to be a smooth
curve-- in other words, that
00:03:12.460 --> 00:03:15.100
'x' is a differentiable
function of 't'.
00:03:15.100 --> 00:03:18.370
Again, in the language of
calculus, what we're saying is
00:03:18.370 --> 00:03:21.340
the slope of this curve exists
at this particular point, and
00:03:21.340 --> 00:03:26.710
it's given by the 'x dt'
evaluated at 't' equals 't1'.
00:03:26.710 --> 00:03:28.980
Now, without going into a proof
at this stage, all we're
00:03:28.980 --> 00:03:30.030
saying is this.
00:03:30.030 --> 00:03:33.990
We suspect that if 'y' is a
differentiable function of
00:03:33.990 --> 00:03:36.930
'x', and 'x' is a differentiable
function of
00:03:36.930 --> 00:03:40.880
't', that therefore, 'y'
should also be a
00:03:40.880 --> 00:03:43.310
differentiable function
of 't'.
00:03:43.310 --> 00:03:46.400
Notice it's not a conjecture
at all that if 'y' is a
00:03:46.400 --> 00:03:49.750
function of 'x' and 'x' is a
function of 't', that 'y' is a
00:03:49.750 --> 00:03:50.870
function of 't'.
00:03:50.870 --> 00:03:52.350
That part is clear.
00:03:52.350 --> 00:03:55.370
The conjecture is that we
suspect that if 'y' is a
00:03:55.370 --> 00:03:58.420
differentiable function
of 'x', and 'x' is a
00:03:58.420 --> 00:04:01.150
differentiable function of
't', that 'y' will be a
00:04:01.150 --> 00:04:03.310
differentiable function
of 't'.
00:04:03.310 --> 00:04:06.870
In still other words, our
suspicion is perhaps that a
00:04:06.870 --> 00:04:10.850
differentiable function of a
differentiable function is
00:04:10.850 --> 00:04:13.490
again a differentiable
function.
00:04:13.490 --> 00:04:17.120
But even more to the point, not
only do we suspect, for
00:04:17.120 --> 00:04:22.250
example, that the 'dy dt' exists
here when 't' equals
00:04:22.250 --> 00:04:27.590
't1', but in line with our
lecture of last time, we might
00:04:27.590 --> 00:04:31.500
even begin to suspect, in
terms of this fractional
00:04:31.500 --> 00:04:35.400
notation, that not only does
the 'dy dt' exist at 't'
00:04:35.400 --> 00:04:39.930
equals 't1', but it can be found
by multiplying 'dy dx'
00:04:39.930 --> 00:04:45.000
evaluated at 'x' equals 'x1' by
the 'x dt' evaluated at 't'
00:04:45.000 --> 00:04:46.060
equals 't1'.
00:04:46.060 --> 00:04:51.190
Again, almost as if the 'dx'
from the numerator here
00:04:51.190 --> 00:04:54.700
canceled with the 'dx' from the
denominator here, the same
00:04:54.700 --> 00:04:56.430
as what we hope our
differential
00:04:56.430 --> 00:04:58.120
notation would be.
00:04:58.120 --> 00:05:01.640
The question is granted that
we would like a result like
00:05:01.640 --> 00:05:05.530
this to hold true, in a course
such as calculus, where we're
00:05:05.530 --> 00:05:08.900
working with very tiny numbers
and quotients of small
00:05:08.900 --> 00:05:12.620
numbers, places where we've seen
that our intuition often
00:05:12.620 --> 00:05:16.390
leads us astray, it becomes
fairly apparent that we had
00:05:16.390 --> 00:05:20.120
better have something stronger
than just intuition in helping
00:05:20.120 --> 00:05:23.580
us derive certain results, no
matter how natural these
00:05:23.580 --> 00:05:25.620
results might look.
00:05:25.620 --> 00:05:28.600
Now, the way we proceed here
is as follows again, and
00:05:28.600 --> 00:05:30.970
notice again the building
blocks of calculus.
00:05:30.970 --> 00:05:35.280
We go back to the fundamental
result of last time.
00:05:35.280 --> 00:05:39.750
You see, after all, to find
'dy dt', we want 'delta y'
00:05:39.750 --> 00:05:42.950
divided by 'delta t', and then
we'll take the limit as 'delta
00:05:42.950 --> 00:05:44.340
t' approaches 0.
00:05:44.340 --> 00:05:47.450
The question is, first of all,
do we have a nice expression
00:05:47.450 --> 00:05:48.880
for 'delta y'?
00:05:48.880 --> 00:05:52.720
And in terms of the lecture of
last time, we saw that if 'y'
00:05:52.720 --> 00:05:56.290
was a differentiable function of
'x', that 'x' equals 'x1',
00:05:56.290 --> 00:06:00.490
that 'delta y' was given by ''dy
dx', evaluated 'x' equals
00:06:00.490 --> 00:06:06.330
'x1' times 'delta x'' plus
'k times delta x'--
00:06:06.330 --> 00:06:09.320
and this is crucial now-- where
the limit of 'k' as
00:06:09.320 --> 00:06:12.270
'delta x' approaches 0 was 0.
00:06:12.270 --> 00:06:16.170
Now you see, this recipe
here is ironclad.
00:06:16.170 --> 00:06:18.780
I emphasized it from a geometric
point of view last
00:06:18.780 --> 00:06:22.250
time, but you may recall that I
proved it from an analytical
00:06:22.250 --> 00:06:22.930
point of view.
00:06:22.930 --> 00:06:25.360
In other words, whether you
want to visualize this or
00:06:25.360 --> 00:06:27.240
derive it, it makes
no difference.
00:06:27.240 --> 00:06:32.030
The key factors that this
statement here is ironclad.
00:06:32.030 --> 00:06:34.710
It's something that we now
know to be true in our
00:06:34.710 --> 00:06:36.510
so-called game of calculus.
00:06:36.510 --> 00:06:41.480
The point is, again, now how do
we use this to check over
00:06:41.480 --> 00:06:43.550
our conjectured result?
00:06:43.550 --> 00:06:46.100
Again, the answer is almost
straightforward.
00:06:46.100 --> 00:06:48.320
If you keep track of these
things, you'll notice that
00:06:48.320 --> 00:06:51.860
calculus is a one-step-at-a-time
procedure.
00:06:51.860 --> 00:06:54.240
Namely, we want 'dy dt'.
00:06:54.240 --> 00:06:57.710
That suggests we first want
'delta y' divided by 'delta
00:06:57.710 --> 00:07:00.340
t', and then we'll take
the limit as 'delta
00:07:00.340 --> 00:07:02.280
t' approaches 0.
00:07:02.280 --> 00:07:04.010
So first, we do this.
00:07:04.010 --> 00:07:07.630
Namely, starting with our
known recipe, we divide
00:07:07.630 --> 00:07:10.300
through by 'delta t', and
why can we do this?
00:07:10.300 --> 00:07:14.120
We can do this because, of
course, 'delta t' is not 0.
00:07:14.120 --> 00:07:17.810
Now we take the limit of both
sides of the equality as
00:07:17.810 --> 00:07:19.630
'delta t' approaches 0.
00:07:19.630 --> 00:07:22.690
We observe that on the left-hand
side, the limit of
00:07:22.690 --> 00:07:26.640
'delta y' divided by 'delta t',
as 'delta t' approaches 0,
00:07:26.640 --> 00:07:30.780
is precisely 'dy dt', and
in this particular case,
00:07:30.780 --> 00:07:33.090
evaluated at 't' equals 't1'.
00:07:33.090 --> 00:07:37.220
In other words, notice that the
left-hand side here, as we
00:07:37.220 --> 00:07:41.420
let 'delta t' approach 0,
becomes the left-hand side of
00:07:41.420 --> 00:07:43.220
our conjecture.
00:07:43.220 --> 00:07:46.070
Now we recall again that the
limit of the sum is the sum of
00:07:46.070 --> 00:07:50.020
the limits, and we now take
the limit of each of these
00:07:50.020 --> 00:07:52.570
terms separately, each
term as a product.
00:07:52.570 --> 00:07:55.080
The limit of a product is the
product of the limits.
00:07:55.080 --> 00:07:58.450
'dy dx' evaluated at 'x' equals
'x1' is a constant.
00:07:58.450 --> 00:08:01.450
In fact, that's just
what, it's 'dy'.
00:08:01.450 --> 00:08:05.950
The limit of 'dy dx' evaluated
'x' equals 'x1', as 'delta t'
00:08:05.950 --> 00:08:11.530
approaches 0, is just 'dy dx'
evaluated 'x' equals 'x1'.
00:08:11.530 --> 00:08:14.080
On the other hand, by
definition, the limit of
00:08:14.080 --> 00:08:18.620
'delta x' divided by 'delta t',
as 'delta t' approaches 0,
00:08:18.620 --> 00:08:21.460
is just 'dx dt'.
00:08:21.460 --> 00:08:25.380
And keeping track of the
subscripts here, later on
00:08:25.380 --> 00:08:28.220
we'll become sloppy and leave
the subscripts out.
00:08:28.220 --> 00:08:30.680
There really is no great
harm done in
00:08:30.680 --> 00:08:32.610
calculus of a single variable.
00:08:32.610 --> 00:08:36.299
We shall find, in calculus of
several variables, that it is
00:08:36.299 --> 00:08:41.980
extremely important to keep
track of the subscripts and
00:08:41.980 --> 00:08:44.730
where the variables are being
evaluated and things of this
00:08:44.730 --> 00:08:45.760
particular type.
00:08:45.760 --> 00:08:48.500
But I just want to get you used
to the fact that these
00:08:48.500 --> 00:08:51.090
are specific numbers that
we're using over here.
00:08:51.090 --> 00:08:54.220
Now let's continue.
00:08:54.220 --> 00:08:57.900
We take the limit of this term
as 'delta t' approaches 0.
00:08:57.900 --> 00:09:04.620
We observe that this becomes 'dx
dt', and the limit of 'k'
00:09:04.620 --> 00:09:06.550
as 'delta t' approaches 0--
00:09:06.550 --> 00:09:09.770
well, as 'delta t' approaches
0, the fact that 'x' is a
00:09:09.770 --> 00:09:12.590
differentiable function of
't' means that 'delta x'
00:09:12.590 --> 00:09:13.820
approaches 0.
00:09:13.820 --> 00:09:19.310
And since the limit of 'k' as
'delta x' approaches 0 is 0,
00:09:19.310 --> 00:09:21.470
this term becomes 0.
00:09:21.470 --> 00:09:26.330
0 times anything is, any
finite number, is 0.
00:09:26.330 --> 00:09:30.500
That means that this term here
in the limit becomes 0, and
00:09:30.500 --> 00:09:35.700
we're left with the
desired result.
00:09:35.700 --> 00:09:39.660
But notice that we did not
arrive at this desired result
00:09:39.660 --> 00:09:40.830
by hand waving.
00:09:40.830 --> 00:09:45.140
We did not say this term 'delta
x' is getting small, so
00:09:45.140 --> 00:09:46.510
it's becoming negligible.
00:09:46.510 --> 00:09:49.330
I can't emphasize this point
enough that it is true that
00:09:49.330 --> 00:09:52.960
'delta x' is becoming small
here, but so is 'delta t', and
00:09:52.960 --> 00:09:58.480
that indicates, essentially,
you're 0 over 0 form.
00:09:58.480 --> 00:10:01.540
And the thing that saves us,
the thing that makes this
00:10:01.540 --> 00:10:05.730
whole term drop out, is the key
fact that 'k' itself goes
00:10:05.730 --> 00:10:08.800
to 0, as 'delta x' goes to 0.
00:10:08.800 --> 00:10:12.390
By the way, there are easier
ways of intuitively trying to
00:10:12.390 --> 00:10:13.830
remember the chain rule.
00:10:13.830 --> 00:10:17.670
For example, one way that people
often try to visualize
00:10:17.670 --> 00:10:19.160
the chain rule is this.
00:10:19.160 --> 00:10:21.930
They'll say, OK, we
want 'dy dt'.
00:10:21.930 --> 00:10:25.100
So let's take 'delta y' divided
by 'delta t', and then
00:10:25.100 --> 00:10:27.700
we'll take the limit as 'delta
t' approaches 0.
00:10:27.700 --> 00:10:30.660
Now, you see in this notation
here, 'delta y' and 'delta t'
00:10:30.660 --> 00:10:32.200
are actually numbers.
00:10:32.200 --> 00:10:36.130
As numbers, we can write these
things in fractional notation,
00:10:36.130 --> 00:10:39.930
and we could write, what, that
'delta y' divided by 'delta t'
00:10:39.930 --> 00:10:44.150
is ''delta y' divided by 'delta
x'' times ''delta x'
00:10:44.150 --> 00:10:45.750
divided by 'delta t''.
00:10:45.750 --> 00:10:49.550
Then we could take the limit,
as delta t approaches 0, and
00:10:49.550 --> 00:10:51.640
we would arrive at
the same result.
00:10:51.640 --> 00:10:55.710
But again, without trying
to make this thing too
00:10:55.710 --> 00:10:59.130
obnoxiously long here, the thing
to keep in mind is that
00:10:59.130 --> 00:11:00.990
'x' is a function of 't'.
00:11:00.990 --> 00:11:03.760
And from a rigorous point of
view, the danger with this
00:11:03.760 --> 00:11:05.230
shortcut technique--
00:11:05.230 --> 00:11:07.920
and it can be patched up but
requires a great deal of
00:11:07.920 --> 00:11:09.420
mathematical analysis--
00:11:09.420 --> 00:11:13.730
the danger here is that as
'delta t' approaches 0, it's
00:11:13.730 --> 00:11:16.530
quite possible that 'delta
x' will be 0.
00:11:16.530 --> 00:11:18.750
In other words, it's possible
that for a given change in
00:11:18.750 --> 00:11:20.810
't', there is no
change in 'x'.
00:11:20.810 --> 00:11:25.030
Now, if 'delta x' happens to
equal 0, then we're in
00:11:25.030 --> 00:11:26.900
trouble over here.
00:11:26.900 --> 00:11:30.560
In other words, in many cases,
this shortened version gives
00:11:30.560 --> 00:11:32.650
us an idea as to what's
going on.
00:11:32.650 --> 00:11:37.610
But our so-called longer method
has no pitfalls to it.
00:11:37.610 --> 00:11:42.210
But enough said for what
this recipe is.
00:11:42.210 --> 00:11:48.570
This result is known as the
chain rule, and this will be
00:11:48.570 --> 00:11:51.640
the topic of the rest
of today's lecture.
00:11:51.640 --> 00:11:53.480
Now, let's take a look at
some of these things
00:11:53.480 --> 00:11:55.580
in a bit more detail.
00:11:55.580 --> 00:11:59.610
For example, let's look
at an illustration.
00:11:59.610 --> 00:12:03.290
Suppose we want to find 'dy dx',
if 'y' is equal to ''x
00:12:03.290 --> 00:12:05.630
squared + 1' squared'.
00:12:05.630 --> 00:12:08.030
Let me first do this problem
the wrong way.
00:12:11.860 --> 00:12:14.570
Let's put a question
mark over here.
00:12:14.570 --> 00:12:18.110
People learn things like, what,
bring the exponent down
00:12:18.110 --> 00:12:22.240
and replace it by one less.
00:12:22.240 --> 00:12:25.350
Now certainly, if I bring the
exponent down here, and
00:12:25.350 --> 00:12:29.302
replace it by one less, this
is the answer that I get.
00:12:29.302 --> 00:12:31.920
Of course the question is,
is this the right answer?
00:12:31.920 --> 00:12:36.620
Well, you see, notice one very
nice way about finding out
00:12:36.620 --> 00:12:39.650
whether an answer is wrong, is
to first find out by another
00:12:39.650 --> 00:12:41.740
way which is the right answer.
00:12:41.740 --> 00:12:45.020
For example, if 'y' equals ''x
squared + 1' squared', it
00:12:45.020 --> 00:12:47.000
happens that we know how
to square this thing.
00:12:47.000 --> 00:12:49.360
We can find directly
that another way of
00:12:49.360 --> 00:12:50.980
expressing 'y' is what?
00:12:50.980 --> 00:12:55.740
It's 'x' to the fourth plus '2x
squared' plus 1, but we
00:12:55.740 --> 00:12:59.800
have previously learned how to
differentiate a polynomial.
00:12:59.800 --> 00:13:01.500
Through the polynomial is what,
this is going to be
00:13:01.500 --> 00:13:08.220
what? '4 x cubed' plus '4x'.
00:13:08.220 --> 00:13:15.170
And you see somehow or other,
this does not seem to give--
00:13:15.170 --> 00:13:16.410
well, for one thing, we
see that these are
00:13:16.410 --> 00:13:17.580
two different answers.
00:13:17.580 --> 00:13:19.630
For another thing, if this is
the one that happens to be the
00:13:19.630 --> 00:13:22.860
right answer, this is the one
that is the wrong answer.
00:13:22.860 --> 00:13:25.490
And since we know from previous
material this is the
00:13:25.490 --> 00:13:28.130
right answer, there is something
wrong with this
00:13:28.130 --> 00:13:30.390
regardless of how right
it might look.
00:13:30.390 --> 00:13:33.200
In fact, how much are
we off by over here?
00:13:37.010 --> 00:13:38.660
If we factor this thing
out, what can we do?
00:13:38.660 --> 00:13:44.700
We can write this as '4x'
times 'x squared + 1'.
00:13:44.700 --> 00:13:47.000
And what we really had
over here was twice
00:13:47.000 --> 00:13:48.420
'x squared + 1'.
00:13:48.420 --> 00:13:52.630
It seems that the correction
factor is '2x'.
00:13:52.630 --> 00:13:55.910
Now again, notice that the
derivative of what's inside
00:13:55.910 --> 00:14:00.360
the parentheses over here just
happens to be exactly '2x'.
00:14:00.360 --> 00:14:03.420
Now how does the chain rule
come into play in a
00:14:03.420 --> 00:14:05.140
problem of this type?
00:14:05.140 --> 00:14:08.810
You see, the thing is, that what
we should do over here is
00:14:08.810 --> 00:14:10.100
rewrite this.
00:14:10.100 --> 00:14:14.970
Namely, for example, let 'u'
equal 'x squared + 1'.
00:14:14.970 --> 00:14:21.150
Then what this says is what? 'y'
is equal to 'u squared',
00:14:21.150 --> 00:14:25.780
where 'u' is equal to
'x squared + 1'.
00:14:25.780 --> 00:14:28.000
This is just another way of
writing this, and in this
00:14:28.000 --> 00:14:30.760
particular form, the
chain rule seems to
00:14:30.760 --> 00:14:32.470
be emphasized more.
00:14:32.470 --> 00:14:36.540
You see, 'y' is a function of
'u', 'u' is a function of 'x'.
00:14:36.540 --> 00:14:39.630
Notice that from the first
equation, it is relatively
00:14:39.630 --> 00:14:42.440
easy to find 'dy du'.
00:14:42.440 --> 00:14:45.760
In fact, it's just what, '2u'.
00:14:45.760 --> 00:14:47.950
We'll write that down later.
00:14:47.950 --> 00:14:51.640
From the second equation, it's
easy to find 'du dx'.
00:14:51.640 --> 00:14:54.940
And by the chain rule, all we're
saying is that 'dy du'
00:14:54.940 --> 00:14:58.140
times 'du dx' is 'dy dx'.
00:14:58.140 --> 00:14:59.960
See, what will that give
us in this case?
00:14:59.960 --> 00:15:02.130
'dy du' is '2u'.
00:15:02.130 --> 00:15:05.610
'du dx' is '2x'.
00:15:05.610 --> 00:15:09.460
That gives us '4x' times 'u'.
00:15:09.460 --> 00:15:16.570
'u' is 'x squared + 1', and so
this becomes '4x' times 'x
00:15:16.570 --> 00:15:18.170
squared + 1'.
00:15:18.170 --> 00:15:21.180
And if we now compare this
with what was the correct
00:15:21.180 --> 00:15:26.250
answer, we see that
in this case,
00:15:26.250 --> 00:15:28.060
everything worked out fine.
00:15:28.060 --> 00:15:31.750
I suppose what we should do here
is to comment now on the
00:15:31.750 --> 00:15:35.280
danger of memorizing recipes
without thoroughly
00:15:35.280 --> 00:15:36.580
understanding them.
00:15:36.580 --> 00:15:39.790
The idea, that said when you
want to differentiate
00:15:39.790 --> 00:15:43.150
something raised to a power that
you bring the power down
00:15:43.150 --> 00:15:47.450
and replace it by one less,
hinged on the fact that the
00:15:47.450 --> 00:15:52.650
thing that was being raised
to the power was the same
00:15:52.650 --> 00:15:55.830
variable with respect to which
you were doing the
00:15:55.830 --> 00:15:57.380
differentiation.
00:15:57.380 --> 00:16:02.780
You see, for example, when we
had 'y' equaled 'x squared',
00:16:02.780 --> 00:16:07.730
and then we wrote that 'dy dx'
is '2x', the thing that was
00:16:07.730 --> 00:16:10.540
important over here was
the fact that what?
00:16:10.540 --> 00:16:13.640
The thing that was being raised
to the second power is
00:16:13.640 --> 00:16:17.320
precisely the variable with
respect to which we were doing
00:16:17.320 --> 00:16:19.180
the differentiation.
00:16:19.180 --> 00:16:24.400
You see, in the problem 'y'
equals 'x squared + 1'
00:16:24.400 --> 00:16:28.220
squared, the thing that was
being raised to the second
00:16:28.220 --> 00:16:31.190
power was 'x squared + 1'.
00:16:31.190 --> 00:16:33.170
The variable with respect
to which we were
00:16:33.170 --> 00:16:36.020
differentiating was 'x'.
00:16:36.020 --> 00:16:40.060
In other words, to write this
thing more symbolically, if
00:16:40.060 --> 00:16:47.730
'y' is equal to something,
square it, then the derivative
00:16:47.730 --> 00:16:51.390
that's equal to twice that
something is the derivative of
00:16:51.390 --> 00:16:54.870
'y' with respect to
that something.
00:16:54.870 --> 00:16:57.560
You see, the place the chain
rule comes in is when the
00:16:57.560 --> 00:17:01.830
variable which appears here,
is not the same as the
00:17:01.830 --> 00:17:06.150
variable which appears here, and
we'll see this in greater
00:17:06.150 --> 00:17:08.390
detail as we go along.
00:17:08.390 --> 00:17:12.339
By the way, the chain rule comes
up in another form known
00:17:12.339 --> 00:17:16.750
as parametric equations, and
this form comes up very often.
00:17:16.750 --> 00:17:19.980
It's a twist of what we were
talking about before.
00:17:19.980 --> 00:17:24.210
This is the situation in which
frequently we want to compare
00:17:24.210 --> 00:17:25.589
two variables.
00:17:25.589 --> 00:17:28.580
Let's call them 'x' and
'y', all right?
00:17:28.580 --> 00:17:32.730
And it happens that both
variables, 'x' and 'y', can be
00:17:32.730 --> 00:17:36.760
expressed more simply in terms
of a third variable, 't'.
00:17:36.760 --> 00:17:40.040
And frequently, what one does
is try to talk about the
00:17:40.040 --> 00:17:43.650
relationship that exists between
'y' and 'x' in terms
00:17:43.650 --> 00:17:47.120
of eliminating t between
these two equations.
00:17:47.120 --> 00:17:50.390
By the way, in terms of
differential language, there
00:17:50.390 --> 00:17:52.960
seems to be an easier way
of handling this.
00:17:52.960 --> 00:17:56.380
Namely, you see, if we
differentiate the first
00:17:56.380 --> 00:18:01.860
equation, we get what, that
'dy dt' is 'f prime of t'.
00:18:01.860 --> 00:18:06.610
If we differentiate the second
equation, we get that 'dx dt'
00:18:06.610 --> 00:18:08.660
is 'g prime of t'.
00:18:08.660 --> 00:18:13.590
Now if, as we said in our last
lecture, we can pretend that
00:18:13.590 --> 00:18:15.460
this is really a fraction,
that it's
00:18:15.460 --> 00:18:17.550
'dy' divided by 'dt'--
00:18:17.550 --> 00:18:21.090
in other words, if we think of
'dy' as being 'delta y-tan',
00:18:21.090 --> 00:18:25.430
of 'dx' as being 'delta x-tan',
and 'dt' as being
00:18:25.430 --> 00:18:29.090
'delta t', it would appear that
we could say, what, that
00:18:29.090 --> 00:18:36.980
'dy dt' divided by 'dx dt'
would just be what?
00:18:36.980 --> 00:18:38.020
'dy dx'.
00:18:38.020 --> 00:18:42.890
In other words, ''dy' divided
by 'dt'' divided by ''dx'
00:18:42.890 --> 00:18:45.600
divided by 'dt'', which is what
this would say if this
00:18:45.600 --> 00:18:49.670
was in differential form,
would just be 'dy dx'.
00:18:49.670 --> 00:18:52.570
In other words, we get the
feeling that to find the
00:18:52.570 --> 00:18:56.580
derivative here, all we have
to do is differentiate 'y'
00:18:56.580 --> 00:18:59.870
with respect to 't', and divide
that by the derivative
00:18:59.870 --> 00:19:01.910
of 'x' with respect to 't'.
00:19:04.440 --> 00:19:07.110
And by the way, you see, this
becomes a particularly
00:19:07.110 --> 00:19:11.590
powerful tool in those
computational cases where we
00:19:11.590 --> 00:19:16.090
do not know how to eliminate
't', and to solve specifically
00:19:16.090 --> 00:19:17.860
for 'y' in terms of 'x'.
00:19:17.860 --> 00:19:20.700
You see, in terms of this
particular recipe over here,
00:19:20.700 --> 00:19:24.890
we are allowed to leave 'x'
and 'y' in terms of 't'.
00:19:24.890 --> 00:19:28.500
Again, the same old bugaboo
comes up to plague us.
00:19:28.500 --> 00:19:32.420
The fact that something seems
natural is not enough to allow
00:19:32.420 --> 00:19:35.060
us to believe that it's
actually correct.
00:19:35.060 --> 00:19:38.930
Is there a more rigorous way of
obtaining the same result?
00:19:38.930 --> 00:19:41.070
Again, the answer is yes.
00:19:41.070 --> 00:19:43.920
And not only is the answer yes,
but it goes back to the
00:19:43.920 --> 00:19:46.370
fundamental recipe that we
were discussing in our
00:19:46.370 --> 00:19:47.690
previous lecture.
00:19:47.690 --> 00:19:52.980
Namely, we know that 'delta y'
is ''f prime of t' times
00:19:52.980 --> 00:20:00.850
'delta t'' plus 'k1 delta t',
and the 'delta x' is ''g prime
00:20:00.850 --> 00:20:06.320
of t' times 'delta t'' plus 'k2
delta t', where both the
00:20:06.320 --> 00:20:13.260
limit of 'k1' and 'k2' as
'delta t' approach 0.
00:20:13.260 --> 00:20:15.490
And this is a notation, I think,
that takes a while to
00:20:15.490 --> 00:20:16.510
get used to.
00:20:16.510 --> 00:20:19.570
We're used to seeing letters
like 'k' stand for constants,
00:20:19.570 --> 00:20:22.650
but it's important over here
to understand that 'k1' and
00:20:22.650 --> 00:20:26.670
'k2' are functions of 'delta
t', that the difference
00:20:26.670 --> 00:20:30.440
between 'delta y' and 'delta
y-tan', 'delta x' and 'delta
00:20:30.440 --> 00:20:35.260
x-tan', that difference, which
is 'k delta x' or 'k delta y',
00:20:35.260 --> 00:20:37.820
depending on which problem
we're dealing with that's
00:20:37.820 --> 00:20:44.670
certainly 'k' in that case, does
depend on how big 'delta
00:20:44.670 --> 00:20:46.030
t' happens to be.
00:20:46.030 --> 00:20:49.430
At any rate, the important thing
is that as 'delta t'
00:20:49.430 --> 00:20:52.430
approaches 0, these
go to 0 also.
00:20:52.430 --> 00:20:56.570
Now you see if we take this, and
actually compute 'delta y'
00:20:56.570 --> 00:20:58.005
divided by 'delta x'--
00:21:01.010 --> 00:21:06.940
and we'll write this a little
bit more suggestively, factor
00:21:06.940 --> 00:21:10.325
out 'delta t' from both
numerator and denominator--
00:21:14.020 --> 00:21:16.740
it rigorously tells us what
'delta y' divided
00:21:16.740 --> 00:21:18.400
by 'delta x' is.
00:21:18.400 --> 00:21:23.890
Now we take the limit, as
'delta x' approaches 0.
00:21:23.890 --> 00:21:26.300
That, by definition, is what?
00:21:26.300 --> 00:21:28.180
That's by definition 'dy dx'.
00:21:28.180 --> 00:21:33.460
Well, you see, first of all,
we cancel out the 'delta t'
00:21:33.460 --> 00:21:37.510
over here, see, 'delta t' is
not 0, we're assuming.
00:21:37.510 --> 00:21:40.370
Since it's not 0 it can be
canceled out, and once we've
00:21:40.370 --> 00:21:43.950
canceled out 'delta t', notice
that as 'delta t' approaches
00:21:43.950 --> 00:21:47.320
0, so does 'delta x'.
00:21:47.320 --> 00:21:49.130
As 'delta x' approaches
0, so does 'delta t'.
00:21:49.130 --> 00:21:52.050
That makes 'k1' and
'k2' go to 0.
00:21:52.050 --> 00:21:54.660
And then since the limit of a
quotient is the quotient of
00:21:54.660 --> 00:21:58.530
the limits, provided only the
'g prime of t' is not 0, we
00:21:58.530 --> 00:22:01.330
see that in the eliminating
process, we
00:22:01.330 --> 00:22:04.150
get the same answer.
00:22:04.150 --> 00:22:07.170
And by the way, see, once we
get the same answer, as we
00:22:07.170 --> 00:22:11.050
would have got the short way,
then we can use the
00:22:11.050 --> 00:22:13.500
convenience of the
short recipe.
00:22:13.500 --> 00:22:17.430
However, the fact that the short
recipe was nice is not
00:22:17.430 --> 00:22:19.620
enough of a guarantee
that it was giving
00:22:19.620 --> 00:22:21.050
us the correct answer.
00:22:21.050 --> 00:22:25.570
As a case in point, it's rather
interesting to point
00:22:25.570 --> 00:22:28.200
out that if you want the
second derivative--
00:22:28.200 --> 00:22:29.940
in other words, let's recall
what we have here.
00:22:29.940 --> 00:22:36.770
We have what? 'y' was given
to 'b', say 'f of t'.
00:22:36.770 --> 00:22:39.960
'x' was given by 'g of t'.
00:22:39.960 --> 00:22:43.100
And you see from these two
equations, what we
00:22:43.100 --> 00:22:44.450
could do is find what?
00:22:44.450 --> 00:22:48.660
We could find the second
derivative of 'y with respect
00:22:48.660 --> 00:22:52.310
to t', and we could also find
from this equation the second
00:22:52.310 --> 00:22:54.450
derivative of 'x' with
respect to 't'.
00:22:54.450 --> 00:22:56.280
This we could certainly do.
00:22:56.280 --> 00:22:59.680
And mechanically, we could
certainly say, let's cancel
00:22:59.680 --> 00:23:01.770
the common denominator.
00:23:01.770 --> 00:23:04.490
The interesting thing is that
when you form that quotient,
00:23:04.490 --> 00:23:08.170
whatever that quotient is, it
does not come out to be the
00:23:08.170 --> 00:23:10.900
second derivative of 'y'
with respect to 'x'.
00:23:10.900 --> 00:23:13.990
And there is an interesting
piece of folklore over here.
00:23:13.990 --> 00:23:17.040
I don't know if this ever
bothered you or not, but it
00:23:17.040 --> 00:23:18.170
used to bother me.
00:23:18.170 --> 00:23:20.720
I never understood why, when
you talk about the second
00:23:20.720 --> 00:23:24.130
derivative, that the exponent
was written between the 'd'
00:23:24.130 --> 00:23:28.965
and the variable in one case,
but written at the end in the
00:23:28.965 --> 00:23:29.270
other case.
00:23:29.270 --> 00:23:32.010
In other words, notice that the
2 here appears between the
00:23:32.010 --> 00:23:34.340
'd' and the 'y', but
in the denominator,
00:23:34.340 --> 00:23:35.990
the 'd' appears outside.
00:23:35.990 --> 00:23:39.510
And again, it was the foresight
of the fathers of
00:23:39.510 --> 00:23:42.950
differential calculus who
noticed rather interestingly
00:23:42.950 --> 00:23:47.190
that if mechanically you did
agree to cancel the common
00:23:47.190 --> 00:23:51.560
denominator here, that what you
would wind up with is not
00:23:51.560 --> 00:23:56.900
'd2y dx squared',
but rather what?
00:23:56.900 --> 00:24:00.860
'd2y d2x'.
00:24:00.860 --> 00:24:03.280
In other words, if you
mechanically carried this out,
00:24:03.280 --> 00:24:06.390
notice that the notation
would be incorrect.
00:24:06.390 --> 00:24:10.390
The 2 comes out to be in the
wrong place over here.
00:24:10.390 --> 00:24:14.750
You see, again, the interesting
point is we don't
00:24:14.750 --> 00:24:17.800
have to rely on taking
my word for it.
00:24:17.800 --> 00:24:21.220
Somebody might say to me, now
look, all you've told me is
00:24:21.220 --> 00:24:25.120
that I get the wrong answer
solving this problem this
00:24:25.120 --> 00:24:26.600
particular way.
00:24:26.600 --> 00:24:29.040
And you've given me a nice
lecture about how the 2's come
00:24:29.040 --> 00:24:30.410
out the wrong way
and everything.
00:24:30.410 --> 00:24:33.456
How do I know that this
is the wrong answer?
00:24:33.456 --> 00:24:35.930
See, and again, everything
comes back to
00:24:35.930 --> 00:24:37.360
fundamentals again.
00:24:37.360 --> 00:24:42.500
To find 'd2y dx squared',
observe that by definition,
00:24:42.500 --> 00:24:46.210
that's just 'd dx' of 'dy dx'.
00:24:49.220 --> 00:24:51.470
That definition doesn't depend
on what functions we're
00:24:51.470 --> 00:24:51.940
dealing with.
00:24:51.940 --> 00:24:54.300
The second derivative with
respect to 'x' is the
00:24:54.300 --> 00:24:57.450
derivative with respect to 'x'
of the first derivative.
00:24:57.450 --> 00:25:00.730
Now, once we have this, you see,
knowing from our previous
00:25:00.730 --> 00:25:03.620
case, that what?
00:25:03.620 --> 00:25:10.550
'dy dx' was 'f prime of t'
divided by 'g prime of t'.
00:25:10.550 --> 00:25:12.580
We can now do what?
00:25:12.580 --> 00:25:13.890
Take this derivative.
00:25:13.890 --> 00:25:16.240
By the way, again, notice
how the chain
00:25:16.240 --> 00:25:18.330
rule comes up in practice.
00:25:18.330 --> 00:25:20.220
It's not always dictated
to us.
00:25:20.220 --> 00:25:24.000
If you look at the expression
inside the parentheses, what
00:25:24.000 --> 00:25:25.110
do we have?
00:25:25.110 --> 00:25:28.400
Inside the parentheses, we have
a function of 't' only.
00:25:28.400 --> 00:25:29.830
This is a function of 't'.
00:25:29.830 --> 00:25:32.780
We want to differentiate
it with respect to 'x'.
00:25:32.780 --> 00:25:35.880
The most natural variable to
differentiate a function of
00:25:35.880 --> 00:25:39.210
't' with respect to
is 't' itself.
00:25:39.210 --> 00:25:42.370
In other words, what would've
been nice is if this was the
00:25:42.370 --> 00:25:47.780
derivative of 'f prime of t'
over 'g prime of t', with
00:25:47.780 --> 00:25:49.970
respect to 't'.
00:25:49.970 --> 00:25:51.760
See, this would be
easier to handle.
00:25:51.760 --> 00:25:53.330
We would then use the quotient
rule, et cetera.
00:25:53.330 --> 00:25:55.930
You see, we can differentiate
a function of 't'
00:25:55.930 --> 00:25:57.380
with respect to 't'.
00:25:57.380 --> 00:25:59.380
The trouble is we have
the derivative
00:25:59.380 --> 00:26:00.570
with respect to 'x'.
00:26:00.570 --> 00:26:03.885
And if we just change this to
a 't', that's cheating.
00:26:03.885 --> 00:26:06.600
See, I mean, you pretend you
copy it wrong, because it's an
00:26:06.600 --> 00:26:08.610
easier problem to
solve that way.
00:26:08.610 --> 00:26:12.010
The beauty of the chain rule is
that it allows us to do the
00:26:12.010 --> 00:26:15.900
problem the easier way, and
to doctor up the resulting
00:26:15.900 --> 00:26:18.300
incorrect answer by
the right answer.
00:26:18.300 --> 00:26:22.320
Namely, you see what we wanted
to wind up with here is what,
00:26:22.320 --> 00:26:24.860
the derivative not with respect
to 't', but with
00:26:24.860 --> 00:26:26.810
respect to 'x'.
00:26:26.810 --> 00:26:30.730
And so, by using the chain
rule, you see we do what?
00:26:30.730 --> 00:26:34.410
We take the derivative with
respect to 't', multiply that
00:26:34.410 --> 00:26:35.520
by 'dt dx'--
00:26:35.520 --> 00:26:38.950
again, mechanically, almost
as if these canceled.
00:26:38.950 --> 00:26:42.320
But this is the way the chain
rule works, and now, you see,
00:26:42.320 --> 00:26:45.450
I can work this out by
the regular quotient
00:26:45.450 --> 00:26:46.890
rule, which says what?
00:26:46.890 --> 00:26:53.460
It's the denominator times the
derivative of the numerator.
00:26:53.460 --> 00:26:55.530
See, and I am differentiating
out respect to 't', the
00:26:55.530 --> 00:27:02.530
natural variable, minus the
numerator times the derivative
00:27:02.530 --> 00:27:08.695
of the denominator over the
square of the denominator.
00:27:12.790 --> 00:27:16.220
Now, that's a mess by itself,
meaning, what,
00:27:16.220 --> 00:27:17.930
computationally, it's
not that obvious.
00:27:17.930 --> 00:27:20.280
I mean, it's quite a bit of work
to do here, and then that
00:27:20.280 --> 00:27:27.370
whole thing must be multiplied
by 'dt dx'.
00:27:27.370 --> 00:27:31.500
And this, you see, is how one
goes around finding the second
00:27:31.500 --> 00:27:34.010
derivative of 'y' with respect
to 'x' in terms
00:27:34.010 --> 00:27:35.620
of parametric equations.
00:27:35.620 --> 00:27:38.630
And more than once, if you're
not careful, you're going to
00:27:38.630 --> 00:27:42.150
find yourself making serious
mistakes, by forgetting to put
00:27:42.150 --> 00:27:44.740
in this factor of 'dt dx'.
00:27:44.740 --> 00:27:48.450
By the way, an interesting
point is that we have not
00:27:48.450 --> 00:27:49.910
computed 'dt dx'.
00:27:49.910 --> 00:27:52.325
We have computed 'dx dt'.
00:27:55.220 --> 00:27:56.680
Let's go back here.
00:27:56.680 --> 00:27:59.310
See, 'x' was 'g of t'.
00:27:59.310 --> 00:28:06.210
So from that, 'dx dt'
is 'g prime of t'.
00:28:06.210 --> 00:28:09.730
And the question is if 'dx dt'
is 'g prime of t', how does
00:28:09.730 --> 00:28:12.170
one find 'dt dx'?
00:28:12.170 --> 00:28:15.140
And again, I think your
intuition is going to tell you
00:28:15.140 --> 00:28:17.490
to just take reciprocals.
00:28:17.490 --> 00:28:21.230
And again, the question is it's
true that this suggests
00:28:21.230 --> 00:28:24.880
taking reciprocals, but how do
we know that we can do this,
00:28:24.880 --> 00:28:27.950
and if we can do this, what
does it really mean?
00:28:27.950 --> 00:28:30.730
You see, what this is leading
into is what's going to be the
00:28:30.730 --> 00:28:34.720
subject of our lecture next
time, called 'Inverse
00:28:34.720 --> 00:28:35.820
Functions'.
00:28:35.820 --> 00:28:38.550
And just to give you a preview
of what that lecture is about,
00:28:38.550 --> 00:28:41.450
and how we work things like
this, let's take a look at
00:28:41.450 --> 00:28:43.790
what we mean by inverse
functions.
00:28:43.790 --> 00:28:46.300
Well, we won't even mention
it in much detail.
00:28:46.300 --> 00:28:49.770
But let's take a look and see
what's going on over here.
00:28:49.770 --> 00:28:51.360
Let's suppose that the first--
00:28:51.360 --> 00:28:55.320
and by the way, I've started to
abandon using the 't' over
00:28:55.320 --> 00:28:56.450
here all the time.
00:28:56.450 --> 00:28:59.310
I think those of us in
engineering work primarily
00:28:59.310 --> 00:29:02.500
keep thinking of 't' as being
time, and you may get the
00:29:02.500 --> 00:29:06.940
mistaken notion that if the
variable isn't time, the thing
00:29:06.940 --> 00:29:09.000
doesn't work this way.
00:29:09.000 --> 00:29:11.870
In most cases, physically, the
variable that we're interested
00:29:11.870 --> 00:29:13.030
in will be time.
00:29:13.030 --> 00:29:15.780
But just for the idea of getting
you used to the fact
00:29:15.780 --> 00:29:17.780
that it makes no difference what
the name of the variable
00:29:17.780 --> 00:29:20.320
is, I've taken the liberty
of writing this slightly
00:29:20.320 --> 00:29:21.290
differently.
00:29:21.290 --> 00:29:24.570
Namely, I now assume that y is
a differentiable function of
00:29:24.570 --> 00:29:26.200
'u', and that 'u' is a
00:29:26.200 --> 00:29:28.130
differentiable function of 'x'.
00:29:28.130 --> 00:29:31.940
By the chain rule, I now know
that 'y' is a differentiable
00:29:31.940 --> 00:29:37.580
function of 'x', and that 'dy
dx' is 'dy du' times 'du dx'.
00:29:37.580 --> 00:29:40.990
The interesting thing here is,
is that there is nothing in
00:29:40.990 --> 00:29:44.080
the statement of the chain rule
that says that the first
00:29:44.080 --> 00:29:47.030
variable in the third that
'x' and 'y' must
00:29:47.030 --> 00:29:48.280
be different variables.
00:29:48.280 --> 00:29:51.820
In fact, it might happen that
'x' and 'y' are synonyms for
00:29:51.820 --> 00:29:53.100
one another.
00:29:53.100 --> 00:29:55.500
If 'x' and 'y' happen
to be synonyms--
00:29:55.500 --> 00:29:57.600
suppose 'x' and 'y'
are synonyms--
00:29:57.600 --> 00:29:59.460
look what happens over here.
00:29:59.460 --> 00:30:04.740
'dy dx' is then just 'dy
dy', which is 1.
00:30:04.740 --> 00:30:05.480
See, let's write that down.
00:30:05.480 --> 00:30:07.140
That's 'dy dy'.
00:30:07.140 --> 00:30:10.410
This would be 'dy du', and
if 'x' is equal to
00:30:10.410 --> 00:30:13.040
'y', this is 'du dy'.
00:30:13.040 --> 00:30:18.700
And if this is equal to 1, and
this is 'dy du', and this is
00:30:18.700 --> 00:30:23.160
'du dy', what does this tell
us about the relationship
00:30:23.160 --> 00:30:26.110
between 'dy du' and 'du dy'?
00:30:26.110 --> 00:30:28.840
It says their product is 1.
00:30:28.840 --> 00:30:32.250
And if the product is 1, that
by definition means that the
00:30:32.250 --> 00:30:35.330
two factors are reciprocals.
00:30:35.330 --> 00:30:38.440
Now, what I want you to observe
over here is what this
00:30:38.440 --> 00:30:39.990
whole thing means.
00:30:39.990 --> 00:30:44.450
Namely, if 'y' happens to equal
'x', do you see what
00:30:44.450 --> 00:30:45.610
this thing says?
00:30:45.610 --> 00:30:48.090
It says that 'y' is a
differentiable function of
00:30:48.090 --> 00:30:50.740
'u', and 'u' in turn is a
00:30:50.740 --> 00:30:53.050
differentiable function of 'y'.
00:30:53.050 --> 00:30:55.880
That's precisely what we
meant when we talked
00:30:55.880 --> 00:30:57.480
about inverse functions.
00:30:57.480 --> 00:31:00.440
We don't know when an inverse
function exists.
00:31:00.440 --> 00:31:03.960
All we're saying is, is that
if 'f inverse' happens to
00:31:03.960 --> 00:31:09.480
exist over here, to find 'du
dy', all we have to do is take
00:31:09.480 --> 00:31:12.560
the reciprocal of 'dy du'.
00:31:12.560 --> 00:31:15.240
Now again, this is
going to be the
00:31:15.240 --> 00:31:17.230
subject of our next lecture.
00:31:17.230 --> 00:31:20.550
All I wanted to do was to make
this aside for the time being.
00:31:20.550 --> 00:31:23.470
What I want to do to complete
today's lecture is to get to
00:31:23.470 --> 00:31:24.910
something more tangible.
00:31:24.910 --> 00:31:27.140
See, now that we've talked about
the chain rule, we've
00:31:27.140 --> 00:31:30.390
talked about inverse functions a
little bit, and talked about
00:31:30.390 --> 00:31:33.100
these things from a highly
theoretical point of view,
00:31:33.100 --> 00:31:34.980
let's go ahead and
try to solve a
00:31:34.980 --> 00:31:36.550
particularly simple problem.
00:31:36.550 --> 00:31:38.830
By particularly simple,
I mean this.
00:31:38.830 --> 00:31:42.230
I have chosen the numbers to
come out in a very, very easy
00:31:42.230 --> 00:31:45.580
way, so we don't get lost
in the maze of details.
00:31:45.580 --> 00:31:48.110
In other words, there was a
danger that we will confuse
00:31:48.110 --> 00:31:51.450
the computational details
with the theory.
00:31:51.450 --> 00:31:53.560
So to emphasize the theory,
I've tried to pick a
00:31:53.560 --> 00:31:55.820
straightforward simple problem,
but let's see how
00:31:55.820 --> 00:31:57.770
this thing works out.
00:31:57.770 --> 00:32:01.570
Let's suppose that we're given
that 'y' is equal to 't to the
00:32:01.570 --> 00:32:06.010
fourth power', and 'x' is
equal to 't squared'.
00:32:06.010 --> 00:32:08.850
What we would like to do-- and
by the way, notice what this
00:32:08.850 --> 00:32:13.050
thing says, a given value of 't'
determines both an 'x' and
00:32:13.050 --> 00:32:17.530
the 'y', so that makes 'x' and
'y' functionally related.
00:32:17.530 --> 00:32:21.200
Notice that from the first
equation, we can find that 'dy
00:32:21.200 --> 00:32:25.350
dt' is '4 t cubed'.
00:32:25.350 --> 00:32:30.900
From the second equation, we can
find that 'dx dt' is '2t'.
00:32:30.900 --> 00:32:36.280
And if we now use the chain
rule, 'dy dx' will be what?
00:32:36.280 --> 00:32:42.580
It'll be 'dy dt' divided
by 'dx dt', and
00:32:42.580 --> 00:32:48.140
that's just '2 t squared'.
00:32:48.140 --> 00:32:52.890
By the way, as a check,
notice this.
00:32:52.890 --> 00:32:56.150
If 'y' is equal to 't to the
fourth', and 'x' is equal to
00:32:56.150 --> 00:32:59.970
't squared', since 't to the
fourth' is the square of 't
00:32:59.970 --> 00:33:04.280
squared', that says 'y' is
equal to ''t squared'
00:33:04.280 --> 00:33:08.900
squared', 'y' is equal
to 'x squared'.
00:33:08.900 --> 00:33:12.180
And if 'y' is equal to 'x
squared', in this case, it's
00:33:12.180 --> 00:33:16.800
very easy to see that 'dy
dx' is equal to '2x'.
00:33:16.800 --> 00:33:20.530
By the way, when we try to
compare these two answers,
00:33:20.530 --> 00:33:22.340
they look different, but
that's because they're
00:33:22.340 --> 00:33:24.750
expressed in terms of
different variables.
00:33:24.750 --> 00:33:30.430
If we return to our original
equations, and we see that 'x'
00:33:30.430 --> 00:33:32.530
is equal to 't squared'--
00:33:32.530 --> 00:33:34.470
'x' is a synonym for
't squared'--
00:33:34.470 --> 00:33:38.080
this is the check that we have
received the right answer.
00:33:38.080 --> 00:33:41.400
By the way, before I conclude
today's lecture, I would like
00:33:41.400 --> 00:33:43.630
to make a rather
important aside
00:33:43.630 --> 00:33:46.060
about parametric equations.
00:33:46.060 --> 00:33:49.500
After one works the problem this
way, and comes down to
00:33:49.500 --> 00:33:53.300
the check, and says, hey, after
all of this mess over
00:33:53.300 --> 00:33:56.610
here, I could have replaced
it by just 'y' equals 'x
00:33:56.610 --> 00:33:59.000
squared', why did I have
to work with this
00:33:59.000 --> 00:34:00.220
in the first place?
00:34:00.220 --> 00:34:02.770
We are going to have many, many
examples throughout the
00:34:02.770 --> 00:34:04.770
course that will illustrate
this.
00:34:04.770 --> 00:34:07.440
But at least once in a lecture,
I would like to go on
00:34:07.440 --> 00:34:12.380
record as pointing out that this
pair of equations tells
00:34:12.380 --> 00:34:15.250
you much more than this
equation here.
00:34:15.250 --> 00:34:17.909
This equation simply
tells you this.
00:34:17.909 --> 00:34:21.880
If a particle were moving along
a curve with respect to
00:34:21.880 --> 00:34:26.139
time according to these
equations, this equation here
00:34:26.139 --> 00:34:29.050
simply tells you what path the
particle would follow.
00:34:29.050 --> 00:34:34.480
Namely, the parabola 'y'
equals 'x squared'.
00:34:34.480 --> 00:34:38.100
On the other hand, these
two equations tell you
00:34:38.100 --> 00:34:39.420
much more than that.
00:34:39.420 --> 00:34:43.719
These not only tell you that the
particle moved along the
00:34:43.719 --> 00:34:47.310
parabola 'y' equals 'x squared',
but rather, it tells
00:34:47.310 --> 00:34:53.020
you at a particular time the
point on the parabola that the
00:34:53.020 --> 00:34:54.630
particle was at.
00:34:54.630 --> 00:34:56.130
What I mean is this.
00:34:56.130 --> 00:35:00.480
As another example, suppose we
had 'y' equals 't squared,'
00:35:00.480 --> 00:35:02.920
and 'x' equals 't'.
00:35:02.920 --> 00:35:07.090
If we eliminate 't' from these
two equations, we also find
00:35:07.090 --> 00:35:09.960
that 'y' is equal
to 'x squared'.
00:35:09.960 --> 00:35:15.050
Yet notice that this is not the
same as our original set
00:35:15.050 --> 00:35:16.040
of equations.
00:35:16.040 --> 00:35:20.340
For example, here, when 't' is
2, when 't' is 2 over here,
00:35:20.340 --> 00:35:24.260
what point are we on as far as
the parabola is concerned?
00:35:24.260 --> 00:35:27.930
When 't' is 2, this is
2, and this is 4.
00:35:27.930 --> 00:35:30.760
That would be the
point 2 comma 4.
00:35:30.760 --> 00:35:33.960
On the other hand, with respect
to this equation, when
00:35:33.960 --> 00:35:42.230
't' is 2, 'x' is 4, and 'y' is
16, you see both of these
00:35:42.230 --> 00:35:47.940
particles would follow the same
curve, but they are at
00:35:47.940 --> 00:35:50.500
different points at
different times.
00:35:50.500 --> 00:35:53.360
So don't belittle the
parametric approach.
00:35:53.360 --> 00:35:56.980
Having the parameter 't' in
there tells you more than just
00:35:56.980 --> 00:35:59.680
what the path of
the motion is.
00:35:59.680 --> 00:36:02.380
It tells you at what time
a particle was at what
00:36:02.380 --> 00:36:03.700
particular point.
00:36:03.700 --> 00:36:05.550
Well, enough about that.
00:36:05.550 --> 00:36:08.870
Let's go ahead and find the
second derivative now.
00:36:08.870 --> 00:36:12.770
You see, we already know that
'dy dx' is '2 t squared'.
00:36:12.770 --> 00:36:18.230
Now what we'd like to find
is 'd2y dx squared'.
00:36:18.230 --> 00:36:20.710
Again, the same basic
definition.
00:36:20.710 --> 00:36:22.260
'd2y dx squared'.
00:36:22.260 --> 00:36:24.930
The second derivative is the
derivative of the first
00:36:24.930 --> 00:36:26.970
derivative.
00:36:26.970 --> 00:36:30.600
The first derivative we saw was
'2 t squared', so this is
00:36:30.600 --> 00:36:33.200
the derivative of
'2 t squared'.
00:36:33.200 --> 00:36:35.470
Again, and this is where most
of the mistakes are made,
00:36:35.470 --> 00:36:37.000
people get sloppy.
00:36:37.000 --> 00:36:38.720
They forget the 'x'
is in here.
00:36:38.720 --> 00:36:41.710
They say I know the derivative
of this, it's '4t'.
00:36:41.710 --> 00:36:45.600
Well, the derivative of this is
'4t' with respect to 't'.
00:36:45.600 --> 00:36:49.560
We want to differentiate
with respect to 'x'.
00:36:49.560 --> 00:36:54.020
And the way the chain rule comes
in, we say OK, since 't'
00:36:54.020 --> 00:36:56.660
is the natural variable with
respect to which to
00:36:56.660 --> 00:36:58.470
differentiate, let's do it.
00:36:58.470 --> 00:37:00.820
We'll differentiate
in respect to 't'.
00:37:00.820 --> 00:37:04.270
But since the final answer has
to be with respect to 'x', our
00:37:04.270 --> 00:37:08.590
correction factor by the chain
rule will be 'dt dx'.
00:37:08.590 --> 00:37:11.710
Well, the derivative of '2 t
squared' with respect to 't'
00:37:11.710 --> 00:37:13.750
is clearly '4t'.
00:37:13.750 --> 00:37:16.860
The derivative of 't' with
respect to 'x', assuming that
00:37:16.860 --> 00:37:19.470
we know something about inverse
functions, that's the
00:37:19.470 --> 00:37:21.680
reciprocal of 'dx dt'.
00:37:21.680 --> 00:37:25.270
We just saw that 'dx
dt' was '2t'.
00:37:25.270 --> 00:37:28.950
Therefore, 'dt dx' is 1 over
'2t', and therefore, the
00:37:28.950 --> 00:37:31.870
correct answer appears
to be 2.
00:37:31.870 --> 00:37:35.330
Again, this is why I picked
the simple case.
00:37:35.330 --> 00:37:38.800
Given that 'y' equals 'x
squared', we see at a glance
00:37:38.800 --> 00:37:42.520
that 'dy dx' is equal to '2x',
and also at a glance,
00:37:42.520 --> 00:37:48.330
therefore, that 'd2y dx squared'
is equal to 2.
00:37:48.330 --> 00:37:51.520
By the way, that's exactly
what this is equal to.
00:37:51.520 --> 00:37:54.660
You see, had we forgot the chain
rule, and had we left
00:37:54.660 --> 00:37:57.610
this factor out, this would
have given us--
00:37:57.610 --> 00:37:59.530
in other words, to simply write
down that the answer was
00:37:59.530 --> 00:38:02.510
'4t', which is the most common
mistake that's made, would
00:38:02.510 --> 00:38:04.300
have given us the
wrong answer.
00:38:04.300 --> 00:38:06.300
That's why I put such
an easy problem.
00:38:06.300 --> 00:38:09.150
You see, if I had picked a
tougher computational problem,
00:38:09.150 --> 00:38:10.940
the theory would have
remained the same.
00:38:10.940 --> 00:38:13.190
But when I got two different
answers, it would have been
00:38:13.190 --> 00:38:17.380
difficult to determine which
was the correct answer, and
00:38:17.380 --> 00:38:19.420
which was the incorrect
answer.
00:38:19.420 --> 00:38:22.140
But again, to summarize today's
lecture, it was a
00:38:22.140 --> 00:38:25.510
continuation in a way of the
lecture of last time, when we
00:38:25.510 --> 00:38:29.370
developed the primary recipe
involving differentials.
00:38:29.370 --> 00:38:31.440
Now we applied that
to find something
00:38:31.440 --> 00:38:33.190
called the chain rule.
00:38:33.190 --> 00:38:37.130
In the process of emphasizing
the chain rule, we talked
00:38:37.130 --> 00:38:39.320
about the necessity of
knowing something
00:38:39.320 --> 00:38:41.200
about inverse functions.
00:38:41.200 --> 00:38:45.440
Consequently, that dictates what
our next lecture will be
00:38:45.440 --> 00:38:49.100
concerned with, namely
inverse functions.
00:38:49.100 --> 00:38:51.390
And so until next
time, goodbye.
00:38:54.730 --> 00:38:57.270
ANNOUNCER: Funding for the
publication of this video was
00:38:57.270 --> 00:39:01.980
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:39:01.980 --> 00:39:06.160
Help OCW continue to provide
free and open access to MIT
00:39:06.160 --> 00:39:10.350
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at ocw.mit.edu/donate.