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PROFESSOR: Hi.
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Welcome once again to our
lectures in Calculus
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Revisited, where today we are
going to talk about the
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calculus of composite
functions.
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00:00:42,350 --> 00:00:46,120
Now, recall that we have already
mentioned in previous
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lectures the notion of
a composite function.
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And what we're going to do today
is to emphasize the idea
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00:00:53,490 --> 00:00:58,250
as to how often we are called
upon to find functional
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00:00:58,250 --> 00:01:01,500
relationships, where the first
variable is given in terms of
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00:01:01,500 --> 00:01:04,730
a second variable, and the
second variable, say, is given
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00:01:04,730 --> 00:01:06,590
in terms of the third
variable.
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00:01:06,590 --> 00:01:09,210
And we wish to find, say,
the first variable in
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terms of the third.
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Fact here is where the name "the
chain rule" seems to come
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from, a chain reaction where the
variables are related in a
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00:01:17,770 --> 00:01:18,940
chain this way.
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00:01:18,940 --> 00:01:22,570
Now, we can see this quite
easily in terms of a diagram.
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00:01:22,570 --> 00:01:25,930
Suppose, for example,
that I have a graph
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00:01:25,930 --> 00:01:28,330
of 'x' versus 'y'.
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00:01:28,330 --> 00:01:33,970
And suppose also, I have a
graph of 't' versus 'x'.
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00:01:33,970 --> 00:01:35,850
Without any reference
to calculus--
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00:01:35,850 --> 00:01:37,210
and this is rather important--
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00:01:37,210 --> 00:01:40,470
without any reference to
calculus, notice that these
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00:01:40,470 --> 00:01:45,010
two graphs together allow
me to visualize
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00:01:45,010 --> 00:01:47,330
'y' in terms of 't'.
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00:01:47,330 --> 00:01:50,430
For example, given a particular
value of 't'--
35
00:01:50,430 --> 00:01:52,210
let's call it 't sub 1'--
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00:01:52,210 --> 00:01:57,570
given a value of 't', from that
value of 't', I can find
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00:01:57,570 --> 00:01:59,710
the corresponding
value of 'x'.
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00:01:59,710 --> 00:02:01,780
Let's call that 'x sub 1'.
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00:02:01,780 --> 00:02:06,330
Now, knowing 'x sub 1', I can
come to this diagram.
40
00:02:06,330 --> 00:02:10,539
Knowing what 'x sub 1' is,
I can find 'y sub 1'.
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00:02:10,539 --> 00:02:14,590
And so you see in this chain of
two diagrams, a particular
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00:02:14,590 --> 00:02:20,670
value of 't' allows me to find
a particular value of 'y'.
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00:02:20,670 --> 00:02:25,560
And in this particular way,
I can visualize 'y' as a
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00:02:25,560 --> 00:02:27,420
function of 't'.
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00:02:27,420 --> 00:02:30,550
And you see at this stage of the
game, there is absolutely
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00:02:30,550 --> 00:02:34,200
no need to have to have any
knowledge of calculus to
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00:02:34,200 --> 00:02:37,960
understand what it is that
we're discussing.
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00:02:37,960 --> 00:02:41,840
The place that calculus comes
in is in the following way.
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00:02:41,840 --> 00:02:45,400
Let's suppose it happened in
this first diagram that the
50
00:02:45,400 --> 00:02:48,290
graph of 'y' versus
'x' was smooth.
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00:02:48,290 --> 00:02:51,430
In other words, let's assume
that 'y' is a differentiable
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00:02:51,430 --> 00:02:52,580
function of 'x'.
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00:02:52,580 --> 00:02:55,690
In particular, the way I've
drawn this diagram here, we're
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00:02:55,690 --> 00:03:01,810
saying suppose 'dy dx' evaluated
at 'x' equals 'x1'
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00:03:01,810 --> 00:03:03,850
happens to exist.
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00:03:03,850 --> 00:03:09,220
And suppose also that this graph
of 'x' versus 't', this
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00:03:09,220 --> 00:03:12,460
also happens to be a smooth
curve-- in other words, that
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'x' is a differentiable
function of 't'.
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00:03:15,100 --> 00:03:18,370
Again, in the language of
calculus, what we're saying is
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00:03:18,370 --> 00:03:21,340
the slope of this curve exists
at this particular point, and
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00:03:21,340 --> 00:03:26,710
it's given by the 'x dt'
evaluated at 't' equals 't1'.
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00:03:26,710 --> 00:03:28,980
Now, without going into a proof
at this stage, all we're
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00:03:28,980 --> 00:03:30,030
saying is this.
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We suspect that if 'y' is a
differentiable function of
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'x', and 'x' is a differentiable
function of
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00:03:36,930 --> 00:03:40,880
't', that therefore, 'y'
should also be a
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00:03:40,880 --> 00:03:43,310
differentiable function
of 't'.
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00:03:43,310 --> 00:03:46,400
Notice it's not a conjecture
at all that if 'y' is a
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00:03:46,400 --> 00:03:49,750
function of 'x' and 'x' is a
function of 't', that 'y' is a
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00:03:49,750 --> 00:03:50,870
function of 't'.
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00:03:50,870 --> 00:03:52,350
That part is clear.
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00:03:52,350 --> 00:03:55,370
The conjecture is that we
suspect that if 'y' is a
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00:03:55,370 --> 00:03:58,420
differentiable function
of 'x', and 'x' is a
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00:03:58,420 --> 00:04:01,150
differentiable function of
't', that 'y' will be a
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00:04:01,150 --> 00:04:03,310
differentiable function
of 't'.
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00:04:03,310 --> 00:04:06,870
In still other words, our
suspicion is perhaps that a
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00:04:06,870 --> 00:04:10,850
differentiable function of a
differentiable function is
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00:04:10,850 --> 00:04:13,490
again a differentiable
function.
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00:04:13,490 --> 00:04:17,120
But even more to the point, not
only do we suspect, for
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00:04:17,120 --> 00:04:22,250
example, that the 'dy dt' exists
here when 't' equals
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00:04:22,250 --> 00:04:27,590
't1', but in line with our
lecture of last time, we might
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00:04:27,590 --> 00:04:31,500
even begin to suspect, in
terms of this fractional
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00:04:31,500 --> 00:04:35,400
notation, that not only does
the 'dy dt' exist at 't'
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00:04:35,400 --> 00:04:39,930
equals 't1', but it can be found
by multiplying 'dy dx'
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00:04:39,930 --> 00:04:45,000
evaluated at 'x' equals 'x1' by
the 'x dt' evaluated at 't'
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00:04:45,000 --> 00:04:46,060
equals 't1'.
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00:04:46,060 --> 00:04:51,190
Again, almost as if the 'dx'
from the numerator here
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00:04:51,190 --> 00:04:54,700
canceled with the 'dx' from the
denominator here, the same
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00:04:54,700 --> 00:04:56,430
as what we hope our
differential
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00:04:56,430 --> 00:04:58,120
notation would be.
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00:04:58,120 --> 00:05:01,640
The question is granted that
we would like a result like
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00:05:01,640 --> 00:05:05,530
this to hold true, in a course
such as calculus, where we're
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00:05:05,530 --> 00:05:08,900
working with very tiny numbers
and quotients of small
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00:05:08,900 --> 00:05:12,620
numbers, places where we've seen
that our intuition often
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00:05:12,620 --> 00:05:16,390
leads us astray, it becomes
fairly apparent that we had
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00:05:16,390 --> 00:05:20,120
better have something stronger
than just intuition in helping
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00:05:20,120 --> 00:05:23,580
us derive certain results, no
matter how natural these
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00:05:23,580 --> 00:05:25,620
results might look.
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00:05:25,620 --> 00:05:28,600
Now, the way we proceed here
is as follows again, and
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00:05:28,600 --> 00:05:30,970
notice again the building
blocks of calculus.
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00:05:30,970 --> 00:05:35,280
We go back to the fundamental
result of last time.
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00:05:35,280 --> 00:05:39,750
You see, after all, to find
'dy dt', we want 'delta y'
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00:05:39,750 --> 00:05:42,950
divided by 'delta t', and then
we'll take the limit as 'delta
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00:05:42,950 --> 00:05:44,340
t' approaches 0.
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00:05:44,340 --> 00:05:47,450
The question is, first of all,
do we have a nice expression
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00:05:47,450 --> 00:05:48,880
for 'delta y'?
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00:05:48,880 --> 00:05:52,720
And in terms of the lecture of
last time, we saw that if 'y'
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00:05:52,720 --> 00:05:56,290
was a differentiable function of
'x', that 'x' equals 'x1',
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00:05:56,290 --> 00:06:00,490
that 'delta y' was given by ''dy
dx', evaluated 'x' equals
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00:06:00,490 --> 00:06:06,330
'x1' times 'delta x'' plus
'k times delta x'--
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00:06:06,330 --> 00:06:09,320
and this is crucial now-- where
the limit of 'k' as
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00:06:09,320 --> 00:06:12,270
'delta x' approaches 0 was 0.
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00:06:12,270 --> 00:06:16,170
Now you see, this recipe
here is ironclad.
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00:06:16,170 --> 00:06:18,780
I emphasized it from a geometric
point of view last
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00:06:18,780 --> 00:06:22,250
time, but you may recall that I
proved it from an analytical
116
00:06:22,250 --> 00:06:22,930
point of view.
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00:06:22,930 --> 00:06:25,360
In other words, whether you
want to visualize this or
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00:06:25,360 --> 00:06:27,240
derive it, it makes
no difference.
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00:06:27,240 --> 00:06:32,030
The key factors that this
statement here is ironclad.
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00:06:32,030 --> 00:06:34,710
It's something that we now
know to be true in our
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00:06:34,710 --> 00:06:36,510
so-called game of calculus.
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00:06:36,510 --> 00:06:41,480
The point is, again, now how do
we use this to check over
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00:06:41,480 --> 00:06:43,550
our conjectured result?
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00:06:43,550 --> 00:06:46,100
Again, the answer is almost
straightforward.
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00:06:46,100 --> 00:06:48,320
If you keep track of these
things, you'll notice that
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00:06:48,320 --> 00:06:51,860
calculus is a one-step-at-a-time
procedure.
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00:06:51,860 --> 00:06:54,240
Namely, we want 'dy dt'.
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00:06:54,240 --> 00:06:57,710
That suggests we first want
'delta y' divided by 'delta
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00:06:57,710 --> 00:07:00,340
t', and then we'll take
the limit as 'delta
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00:07:00,340 --> 00:07:02,280
t' approaches 0.
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00:07:02,280 --> 00:07:04,010
So first, we do this.
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00:07:04,010 --> 00:07:07,630
Namely, starting with our
known recipe, we divide
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00:07:07,630 --> 00:07:10,300
through by 'delta t', and
why can we do this?
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00:07:10,300 --> 00:07:14,120
We can do this because, of
course, 'delta t' is not 0.
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00:07:14,120 --> 00:07:17,810
Now we take the limit of both
sides of the equality as
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00:07:17,810 --> 00:07:19,630
'delta t' approaches 0.
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00:07:19,630 --> 00:07:22,690
We observe that on the left-hand
side, the limit of
138
00:07:22,690 --> 00:07:26,640
'delta y' divided by 'delta t',
as 'delta t' approaches 0,
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00:07:26,640 --> 00:07:30,780
is precisely 'dy dt', and
in this particular case,
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00:07:30,780 --> 00:07:33,090
evaluated at 't' equals 't1'.
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00:07:33,090 --> 00:07:37,220
In other words, notice that the
left-hand side here, as we
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00:07:37,220 --> 00:07:41,420
let 'delta t' approach 0,
becomes the left-hand side of
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00:07:41,420 --> 00:07:43,220
our conjecture.
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00:07:43,220 --> 00:07:46,070
Now we recall again that the
limit of the sum is the sum of
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00:07:46,070 --> 00:07:50,020
the limits, and we now take
the limit of each of these
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00:07:50,020 --> 00:07:52,570
terms separately, each
term as a product.
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00:07:52,570 --> 00:07:55,080
The limit of a product is the
product of the limits.
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00:07:55,080 --> 00:07:58,450
'dy dx' evaluated at 'x' equals
'x1' is a constant.
149
00:07:58,450 --> 00:08:01,450
In fact, that's just
what, it's 'dy'.
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00:08:01,450 --> 00:08:05,950
The limit of 'dy dx' evaluated
'x' equals 'x1', as 'delta t'
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00:08:05,950 --> 00:08:11,530
approaches 0, is just 'dy dx'
evaluated 'x' equals 'x1'.
152
00:08:11,530 --> 00:08:14,080
On the other hand, by
definition, the limit of
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00:08:14,080 --> 00:08:18,620
'delta x' divided by 'delta t',
as 'delta t' approaches 0,
154
00:08:18,620 --> 00:08:21,460
is just 'dx dt'.
155
00:08:21,460 --> 00:08:25,380
And keeping track of the
subscripts here, later on
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00:08:25,380 --> 00:08:28,220
we'll become sloppy and leave
the subscripts out.
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00:08:28,220 --> 00:08:30,680
There really is no great
harm done in
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00:08:30,680 --> 00:08:32,610
calculus of a single variable.
159
00:08:32,610 --> 00:08:36,299
We shall find, in calculus of
several variables, that it is
160
00:08:36,299 --> 00:08:41,980
extremely important to keep
track of the subscripts and
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00:08:41,980 --> 00:08:44,730
where the variables are being
evaluated and things of this
162
00:08:44,730 --> 00:08:45,760
particular type.
163
00:08:45,760 --> 00:08:48,500
But I just want to get you used
to the fact that these
164
00:08:48,500 --> 00:08:51,090
are specific numbers that
we're using over here.
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00:08:51,090 --> 00:08:54,220
Now let's continue.
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00:08:54,220 --> 00:08:57,900
We take the limit of this term
as 'delta t' approaches 0.
167
00:08:57,900 --> 00:09:04,620
We observe that this becomes 'dx
dt', and the limit of 'k'
168
00:09:04,620 --> 00:09:06,550
as 'delta t' approaches 0--
169
00:09:06,550 --> 00:09:09,770
well, as 'delta t' approaches
0, the fact that 'x' is a
170
00:09:09,770 --> 00:09:12,590
differentiable function of
't' means that 'delta x'
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00:09:12,590 --> 00:09:13,820
approaches 0.
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00:09:13,820 --> 00:09:19,310
And since the limit of 'k' as
'delta x' approaches 0 is 0,
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00:09:19,310 --> 00:09:21,470
this term becomes 0.
174
00:09:21,470 --> 00:09:26,330
0 times anything is, any
finite number, is 0.
175
00:09:26,330 --> 00:09:30,500
That means that this term here
in the limit becomes 0, and
176
00:09:30,500 --> 00:09:35,700
we're left with the
desired result.
177
00:09:35,700 --> 00:09:39,660
But notice that we did not
arrive at this desired result
178
00:09:39,660 --> 00:09:40,830
by hand waving.
179
00:09:40,830 --> 00:09:45,140
We did not say this term 'delta
x' is getting small, so
180
00:09:45,140 --> 00:09:46,510
it's becoming negligible.
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00:09:46,510 --> 00:09:49,330
I can't emphasize this point
enough that it is true that
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00:09:49,330 --> 00:09:52,960
'delta x' is becoming small
here, but so is 'delta t', and
183
00:09:52,960 --> 00:09:58,480
that indicates, essentially,
you're 0 over 0 form.
184
00:09:58,480 --> 00:10:01,540
And the thing that saves us,
the thing that makes this
185
00:10:01,540 --> 00:10:05,730
whole term drop out, is the key
fact that 'k' itself goes
186
00:10:05,730 --> 00:10:08,800
to 0, as 'delta x' goes to 0.
187
00:10:08,800 --> 00:10:12,390
By the way, there are easier
ways of intuitively trying to
188
00:10:12,390 --> 00:10:13,830
remember the chain rule.
189
00:10:13,830 --> 00:10:17,670
For example, one way that people
often try to visualize
190
00:10:17,670 --> 00:10:19,160
the chain rule is this.
191
00:10:19,160 --> 00:10:21,930
They'll say, OK, we
want 'dy dt'.
192
00:10:21,930 --> 00:10:25,100
So let's take 'delta y' divided
by 'delta t', and then
193
00:10:25,100 --> 00:10:27,700
we'll take the limit as 'delta
t' approaches 0.
194
00:10:27,700 --> 00:10:30,660
Now, you see in this notation
here, 'delta y' and 'delta t'
195
00:10:30,660 --> 00:10:32,200
are actually numbers.
196
00:10:32,200 --> 00:10:36,130
As numbers, we can write these
things in fractional notation,
197
00:10:36,130 --> 00:10:39,930
and we could write, what, that
'delta y' divided by 'delta t'
198
00:10:39,930 --> 00:10:44,150
is ''delta y' divided by 'delta
x'' times ''delta x'
199
00:10:44,150 --> 00:10:45,750
divided by 'delta t''.
200
00:10:45,750 --> 00:10:49,550
Then we could take the limit,
as delta t approaches 0, and
201
00:10:49,550 --> 00:10:51,640
we would arrive at
the same result.
202
00:10:51,640 --> 00:10:55,710
But again, without trying
to make this thing too
203
00:10:55,710 --> 00:10:59,130
obnoxiously long here, the thing
to keep in mind is that
204
00:10:59,130 --> 00:11:00,990
'x' is a function of 't'.
205
00:11:00,990 --> 00:11:03,760
And from a rigorous point of
view, the danger with this
206
00:11:03,760 --> 00:11:05,230
shortcut technique--
207
00:11:05,230 --> 00:11:07,920
and it can be patched up but
requires a great deal of
208
00:11:07,920 --> 00:11:09,420
mathematical analysis--
209
00:11:09,420 --> 00:11:13,730
the danger here is that as
'delta t' approaches 0, it's
210
00:11:13,730 --> 00:11:16,530
quite possible that 'delta
x' will be 0.
211
00:11:16,530 --> 00:11:18,750
In other words, it's possible
that for a given change in
212
00:11:18,750 --> 00:11:20,810
't', there is no
change in 'x'.
213
00:11:20,810 --> 00:11:25,030
Now, if 'delta x' happens to
equal 0, then we're in
214
00:11:25,030 --> 00:11:26,900
trouble over here.
215
00:11:26,900 --> 00:11:30,560
In other words, in many cases,
this shortened version gives
216
00:11:30,560 --> 00:11:32,650
us an idea as to what's
going on.
217
00:11:32,650 --> 00:11:37,610
But our so-called longer method
has no pitfalls to it.
218
00:11:37,610 --> 00:11:42,210
But enough said for what
this recipe is.
219
00:11:42,210 --> 00:11:48,570
This result is known as the
chain rule, and this will be
220
00:11:48,570 --> 00:11:51,640
the topic of the rest
of today's lecture.
221
00:11:51,640 --> 00:11:53,480
Now, let's take a look at
some of these things
222
00:11:53,480 --> 00:11:55,580
in a bit more detail.
223
00:11:55,580 --> 00:11:59,610
For example, let's look
at an illustration.
224
00:11:59,610 --> 00:12:03,290
Suppose we want to find 'dy dx',
if 'y' is equal to ''x
225
00:12:03,290 --> 00:12:05,630
squared + 1' squared'.
226
00:12:05,630 --> 00:12:08,030
Let me first do this problem
the wrong way.
227
00:12:08,030 --> 00:12:11,860
228
00:12:11,860 --> 00:12:14,570
Let's put a question
mark over here.
229
00:12:14,570 --> 00:12:18,110
People learn things like, what,
bring the exponent down
230
00:12:18,110 --> 00:12:22,240
and replace it by one less.
231
00:12:22,240 --> 00:12:25,350
Now certainly, if I bring the
exponent down here, and
232
00:12:25,350 --> 00:12:29,302
replace it by one less, this
is the answer that I get.
233
00:12:29,302 --> 00:12:31,920
Of course the question is,
is this the right answer?
234
00:12:31,920 --> 00:12:36,620
Well, you see, notice one very
nice way about finding out
235
00:12:36,620 --> 00:12:39,650
whether an answer is wrong, is
to first find out by another
236
00:12:39,650 --> 00:12:41,740
way which is the right answer.
237
00:12:41,740 --> 00:12:45,020
For example, if 'y' equals ''x
squared + 1' squared', it
238
00:12:45,020 --> 00:12:47,000
happens that we know how
to square this thing.
239
00:12:47,000 --> 00:12:49,360
We can find directly
that another way of
240
00:12:49,360 --> 00:12:50,980
expressing 'y' is what?
241
00:12:50,980 --> 00:12:55,740
It's 'x' to the fourth plus '2x
squared' plus 1, but we
242
00:12:55,740 --> 00:12:59,800
have previously learned how to
differentiate a polynomial.
243
00:12:59,800 --> 00:13:01,500
Through the polynomial is what,
this is going to be
244
00:13:01,500 --> 00:13:08,220
what? '4 x cubed' plus '4x'.
245
00:13:08,220 --> 00:13:15,170
And you see somehow or other,
this does not seem to give--
246
00:13:15,170 --> 00:13:16,410
well, for one thing, we
see that these are
247
00:13:16,410 --> 00:13:17,580
two different answers.
248
00:13:17,580 --> 00:13:19,630
For another thing, if this is
the one that happens to be the
249
00:13:19,630 --> 00:13:22,860
right answer, this is the one
that is the wrong answer.
250
00:13:22,860 --> 00:13:25,490
And since we know from previous
material this is the
251
00:13:25,490 --> 00:13:28,130
right answer, there is something
wrong with this
252
00:13:28,130 --> 00:13:30,390
regardless of how right
it might look.
253
00:13:30,390 --> 00:13:33,200
In fact, how much are
we off by over here?
254
00:13:33,200 --> 00:13:37,010
255
00:13:37,010 --> 00:13:38,660
If we factor this thing
out, what can we do?
256
00:13:38,660 --> 00:13:44,700
We can write this as '4x'
times 'x squared + 1'.
257
00:13:44,700 --> 00:13:47,000
And what we really had
over here was twice
258
00:13:47,000 --> 00:13:48,420
'x squared + 1'.
259
00:13:48,420 --> 00:13:52,630
It seems that the correction
factor is '2x'.
260
00:13:52,630 --> 00:13:55,910
Now again, notice that the
derivative of what's inside
261
00:13:55,910 --> 00:14:00,360
the parentheses over here just
happens to be exactly '2x'.
262
00:14:00,360 --> 00:14:03,420
Now how does the chain rule
come into play in a
263
00:14:03,420 --> 00:14:05,140
problem of this type?
264
00:14:05,140 --> 00:14:08,810
You see, the thing is, that what
we should do over here is
265
00:14:08,810 --> 00:14:10,100
rewrite this.
266
00:14:10,100 --> 00:14:14,970
Namely, for example, let 'u'
equal 'x squared + 1'.
267
00:14:14,970 --> 00:14:21,150
Then what this says is what? 'y'
is equal to 'u squared',
268
00:14:21,150 --> 00:14:25,780
where 'u' is equal to
'x squared + 1'.
269
00:14:25,780 --> 00:14:28,000
This is just another way of
writing this, and in this
270
00:14:28,000 --> 00:14:30,760
particular form, the
chain rule seems to
271
00:14:30,760 --> 00:14:32,470
be emphasized more.
272
00:14:32,470 --> 00:14:36,540
You see, 'y' is a function of
'u', 'u' is a function of 'x'.
273
00:14:36,540 --> 00:14:39,630
Notice that from the first
equation, it is relatively
274
00:14:39,630 --> 00:14:42,440
easy to find 'dy du'.
275
00:14:42,440 --> 00:14:45,760
In fact, it's just what, '2u'.
276
00:14:45,760 --> 00:14:47,950
We'll write that down later.
277
00:14:47,950 --> 00:14:51,640
From the second equation, it's
easy to find 'du dx'.
278
00:14:51,640 --> 00:14:54,940
And by the chain rule, all we're
saying is that 'dy du'
279
00:14:54,940 --> 00:14:58,140
times 'du dx' is 'dy dx'.
280
00:14:58,140 --> 00:14:59,960
See, what will that give
us in this case?
281
00:14:59,960 --> 00:15:02,130
'dy du' is '2u'.
282
00:15:02,130 --> 00:15:05,610
'du dx' is '2x'.
283
00:15:05,610 --> 00:15:09,460
That gives us '4x' times 'u'.
284
00:15:09,460 --> 00:15:16,570
'u' is 'x squared + 1', and so
this becomes '4x' times 'x
285
00:15:16,570 --> 00:15:18,170
squared + 1'.
286
00:15:18,170 --> 00:15:21,180
And if we now compare this
with what was the correct
287
00:15:21,180 --> 00:15:26,250
answer, we see that
in this case,
288
00:15:26,250 --> 00:15:28,060
everything worked out fine.
289
00:15:28,060 --> 00:15:31,750
I suppose what we should do here
is to comment now on the
290
00:15:31,750 --> 00:15:35,280
danger of memorizing recipes
without thoroughly
291
00:15:35,280 --> 00:15:36,580
understanding them.
292
00:15:36,580 --> 00:15:39,790
The idea, that said when you
want to differentiate
293
00:15:39,790 --> 00:15:43,150
something raised to a power that
you bring the power down
294
00:15:43,150 --> 00:15:47,450
and replace it by one less,
hinged on the fact that the
295
00:15:47,450 --> 00:15:52,650
thing that was being raised
to the power was the same
296
00:15:52,650 --> 00:15:55,830
variable with respect to which
you were doing the
297
00:15:55,830 --> 00:15:57,380
differentiation.
298
00:15:57,380 --> 00:16:02,780
You see, for example, when we
had 'y' equaled 'x squared',
299
00:16:02,780 --> 00:16:07,730
and then we wrote that 'dy dx'
is '2x', the thing that was
300
00:16:07,730 --> 00:16:10,540
important over here was
the fact that what?
301
00:16:10,540 --> 00:16:13,640
The thing that was being raised
to the second power is
302
00:16:13,640 --> 00:16:17,320
precisely the variable with
respect to which we were doing
303
00:16:17,320 --> 00:16:19,180
the differentiation.
304
00:16:19,180 --> 00:16:24,400
You see, in the problem 'y'
equals 'x squared + 1'
305
00:16:24,400 --> 00:16:28,220
squared, the thing that was
being raised to the second
306
00:16:28,220 --> 00:16:31,190
power was 'x squared + 1'.
307
00:16:31,190 --> 00:16:33,170
The variable with respect
to which we were
308
00:16:33,170 --> 00:16:36,020
differentiating was 'x'.
309
00:16:36,020 --> 00:16:40,060
In other words, to write this
thing more symbolically, if
310
00:16:40,060 --> 00:16:47,730
'y' is equal to something,
square it, then the derivative
311
00:16:47,730 --> 00:16:51,390
that's equal to twice that
something is the derivative of
312
00:16:51,390 --> 00:16:54,870
'y' with respect to
that something.
313
00:16:54,870 --> 00:16:57,560
You see, the place the chain
rule comes in is when the
314
00:16:57,560 --> 00:17:01,830
variable which appears here,
is not the same as the
315
00:17:01,830 --> 00:17:06,150
variable which appears here, and
we'll see this in greater
316
00:17:06,150 --> 00:17:08,390
detail as we go along.
317
00:17:08,390 --> 00:17:12,339
By the way, the chain rule comes
up in another form known
318
00:17:12,339 --> 00:17:16,750
as parametric equations, and
this form comes up very often.
319
00:17:16,750 --> 00:17:19,980
It's a twist of what we were
talking about before.
320
00:17:19,980 --> 00:17:24,210
This is the situation in which
frequently we want to compare
321
00:17:24,210 --> 00:17:25,589
two variables.
322
00:17:25,589 --> 00:17:28,580
Let's call them 'x' and
'y', all right?
323
00:17:28,580 --> 00:17:32,730
And it happens that both
variables, 'x' and 'y', can be
324
00:17:32,730 --> 00:17:36,760
expressed more simply in terms
of a third variable, 't'.
325
00:17:36,760 --> 00:17:40,040
And frequently, what one does
is try to talk about the
326
00:17:40,040 --> 00:17:43,650
relationship that exists between
'y' and 'x' in terms
327
00:17:43,650 --> 00:17:47,120
of eliminating t between
these two equations.
328
00:17:47,120 --> 00:17:50,390
By the way, in terms of
differential language, there
329
00:17:50,390 --> 00:17:52,960
seems to be an easier way
of handling this.
330
00:17:52,960 --> 00:17:56,380
Namely, you see, if we
differentiate the first
331
00:17:56,380 --> 00:18:01,860
equation, we get what, that
'dy dt' is 'f prime of t'.
332
00:18:01,860 --> 00:18:06,610
If we differentiate the second
equation, we get that 'dx dt'
333
00:18:06,610 --> 00:18:08,660
is 'g prime of t'.
334
00:18:08,660 --> 00:18:13,590
Now if, as we said in our last
lecture, we can pretend that
335
00:18:13,590 --> 00:18:15,460
this is really a fraction,
that it's
336
00:18:15,460 --> 00:18:17,550
'dy' divided by 'dt'--
337
00:18:17,550 --> 00:18:21,090
in other words, if we think of
'dy' as being 'delta y-tan',
338
00:18:21,090 --> 00:18:25,430
of 'dx' as being 'delta x-tan',
and 'dt' as being
339
00:18:25,430 --> 00:18:29,090
'delta t', it would appear that
we could say, what, that
340
00:18:29,090 --> 00:18:36,980
'dy dt' divided by 'dx dt'
would just be what?
341
00:18:36,980 --> 00:18:38,020
'dy dx'.
342
00:18:38,020 --> 00:18:42,890
In other words, ''dy' divided
by 'dt'' divided by ''dx'
343
00:18:42,890 --> 00:18:45,600
divided by 'dt'', which is what
this would say if this
344
00:18:45,600 --> 00:18:49,670
was in differential form,
would just be 'dy dx'.
345
00:18:49,670 --> 00:18:52,570
In other words, we get the
feeling that to find the
346
00:18:52,570 --> 00:18:56,580
derivative here, all we have
to do is differentiate 'y'
347
00:18:56,580 --> 00:18:59,870
with respect to 't', and divide
that by the derivative
348
00:18:59,870 --> 00:19:01,910
of 'x' with respect to 't'.
349
00:19:01,910 --> 00:19:04,440
350
00:19:04,440 --> 00:19:07,110
And by the way, you see, this
becomes a particularly
351
00:19:07,110 --> 00:19:11,590
powerful tool in those
computational cases where we
352
00:19:11,590 --> 00:19:16,090
do not know how to eliminate
't', and to solve specifically
353
00:19:16,090 --> 00:19:17,860
for 'y' in terms of 'x'.
354
00:19:17,860 --> 00:19:20,700
You see, in terms of this
particular recipe over here,
355
00:19:20,700 --> 00:19:24,890
we are allowed to leave 'x'
and 'y' in terms of 't'.
356
00:19:24,890 --> 00:19:28,500
Again, the same old bugaboo
comes up to plague us.
357
00:19:28,500 --> 00:19:32,420
The fact that something seems
natural is not enough to allow
358
00:19:32,420 --> 00:19:35,060
us to believe that it's
actually correct.
359
00:19:35,060 --> 00:19:38,930
Is there a more rigorous way of
obtaining the same result?
360
00:19:38,930 --> 00:19:41,070
Again, the answer is yes.
361
00:19:41,070 --> 00:19:43,920
And not only is the answer yes,
but it goes back to the
362
00:19:43,920 --> 00:19:46,370
fundamental recipe that we
were discussing in our
363
00:19:46,370 --> 00:19:47,690
previous lecture.
364
00:19:47,690 --> 00:19:52,980
Namely, we know that 'delta y'
is ''f prime of t' times
365
00:19:52,980 --> 00:20:00,850
'delta t'' plus 'k1 delta t',
and the 'delta x' is ''g prime
366
00:20:00,850 --> 00:20:06,320
of t' times 'delta t'' plus 'k2
delta t', where both the
367
00:20:06,320 --> 00:20:13,260
limit of 'k1' and 'k2' as
'delta t' approach 0.
368
00:20:13,260 --> 00:20:15,490
And this is a notation, I think,
that takes a while to
369
00:20:15,490 --> 00:20:16,510
get used to.
370
00:20:16,510 --> 00:20:19,570
We're used to seeing letters
like 'k' stand for constants,
371
00:20:19,570 --> 00:20:22,650
but it's important over here
to understand that 'k1' and
372
00:20:22,650 --> 00:20:26,670
'k2' are functions of 'delta
t', that the difference
373
00:20:26,670 --> 00:20:30,440
between 'delta y' and 'delta
y-tan', 'delta x' and 'delta
374
00:20:30,440 --> 00:20:35,260
x-tan', that difference, which
is 'k delta x' or 'k delta y',
375
00:20:35,260 --> 00:20:37,820
depending on which problem
we're dealing with that's
376
00:20:37,820 --> 00:20:44,670
certainly 'k' in that case, does
depend on how big 'delta
377
00:20:44,670 --> 00:20:46,030
t' happens to be.
378
00:20:46,030 --> 00:20:49,430
At any rate, the important thing
is that as 'delta t'
379
00:20:49,430 --> 00:20:52,430
approaches 0, these
go to 0 also.
380
00:20:52,430 --> 00:20:56,570
Now you see if we take this, and
actually compute 'delta y'
381
00:20:56,570 --> 00:20:58,005
divided by 'delta x'--
382
00:20:58,005 --> 00:21:01,010
383
00:21:01,010 --> 00:21:06,940
and we'll write this a little
bit more suggestively, factor
384
00:21:06,940 --> 00:21:10,325
out 'delta t' from both
numerator and denominator--
385
00:21:10,325 --> 00:21:14,020
386
00:21:14,020 --> 00:21:16,740
it rigorously tells us what
'delta y' divided
387
00:21:16,740 --> 00:21:18,400
by 'delta x' is.
388
00:21:18,400 --> 00:21:23,890
Now we take the limit, as
'delta x' approaches 0.
389
00:21:23,890 --> 00:21:26,300
That, by definition, is what?
390
00:21:26,300 --> 00:21:28,180
That's by definition 'dy dx'.
391
00:21:28,180 --> 00:21:33,460
Well, you see, first of all,
we cancel out the 'delta t'
392
00:21:33,460 --> 00:21:37,510
over here, see, 'delta t' is
not 0, we're assuming.
393
00:21:37,510 --> 00:21:40,370
Since it's not 0 it can be
canceled out, and once we've
394
00:21:40,370 --> 00:21:43,950
canceled out 'delta t', notice
that as 'delta t' approaches
395
00:21:43,950 --> 00:21:47,320
0, so does 'delta x'.
396
00:21:47,320 --> 00:21:49,130
As 'delta x' approaches
0, so does 'delta t'.
397
00:21:49,130 --> 00:21:52,050
That makes 'k1' and
'k2' go to 0.
398
00:21:52,050 --> 00:21:54,660
And then since the limit of a
quotient is the quotient of
399
00:21:54,660 --> 00:21:58,530
the limits, provided only the
'g prime of t' is not 0, we
400
00:21:58,530 --> 00:22:01,330
see that in the eliminating
process, we
401
00:22:01,330 --> 00:22:04,150
get the same answer.
402
00:22:04,150 --> 00:22:07,170
And by the way, see, once we
get the same answer, as we
403
00:22:07,170 --> 00:22:11,050
would have got the short way,
then we can use the
404
00:22:11,050 --> 00:22:13,500
convenience of the
short recipe.
405
00:22:13,500 --> 00:22:17,430
However, the fact that the short
recipe was nice is not
406
00:22:17,430 --> 00:22:19,620
enough of a guarantee
that it was giving
407
00:22:19,620 --> 00:22:21,050
us the correct answer.
408
00:22:21,050 --> 00:22:25,570
As a case in point, it's rather
interesting to point
409
00:22:25,570 --> 00:22:28,200
out that if you want the
second derivative--
410
00:22:28,200 --> 00:22:29,940
in other words, let's recall
what we have here.
411
00:22:29,940 --> 00:22:36,770
We have what? 'y' was given
to 'b', say 'f of t'.
412
00:22:36,770 --> 00:22:39,960
'x' was given by 'g of t'.
413
00:22:39,960 --> 00:22:43,100
And you see from these two
equations, what we
414
00:22:43,100 --> 00:22:44,450
could do is find what?
415
00:22:44,450 --> 00:22:48,660
We could find the second
derivative of 'y with respect
416
00:22:48,660 --> 00:22:52,310
to t', and we could also find
from this equation the second
417
00:22:52,310 --> 00:22:54,450
derivative of 'x' with
respect to 't'.
418
00:22:54,450 --> 00:22:56,280
This we could certainly do.
419
00:22:56,280 --> 00:22:59,680
And mechanically, we could
certainly say, let's cancel
420
00:22:59,680 --> 00:23:01,770
the common denominator.
421
00:23:01,770 --> 00:23:04,490
The interesting thing is that
when you form that quotient,
422
00:23:04,490 --> 00:23:08,170
whatever that quotient is, it
does not come out to be the
423
00:23:08,170 --> 00:23:10,900
second derivative of 'y'
with respect to 'x'.
424
00:23:10,900 --> 00:23:13,990
And there is an interesting
piece of folklore over here.
425
00:23:13,990 --> 00:23:17,040
I don't know if this ever
bothered you or not, but it
426
00:23:17,040 --> 00:23:18,170
used to bother me.
427
00:23:18,170 --> 00:23:20,720
I never understood why, when
you talk about the second
428
00:23:20,720 --> 00:23:24,130
derivative, that the exponent
was written between the 'd'
429
00:23:24,130 --> 00:23:28,965
and the variable in one case,
but written at the end in the
430
00:23:28,965 --> 00:23:29,270
other case.
431
00:23:29,270 --> 00:23:32,010
In other words, notice that the
2 here appears between the
432
00:23:32,010 --> 00:23:34,340
'd' and the 'y', but
in the denominator,
433
00:23:34,340 --> 00:23:35,990
the 'd' appears outside.
434
00:23:35,990 --> 00:23:39,510
And again, it was the foresight
of the fathers of
435
00:23:39,510 --> 00:23:42,950
differential calculus who
noticed rather interestingly
436
00:23:42,950 --> 00:23:47,190
that if mechanically you did
agree to cancel the common
437
00:23:47,190 --> 00:23:51,560
denominator here, that what you
would wind up with is not
438
00:23:51,560 --> 00:23:56,900
'd2y dx squared',
but rather what?
439
00:23:56,900 --> 00:24:00,860
'd2y d2x'.
440
00:24:00,860 --> 00:24:03,280
In other words, if you
mechanically carried this out,
441
00:24:03,280 --> 00:24:06,390
notice that the notation
would be incorrect.
442
00:24:06,390 --> 00:24:10,390
The 2 comes out to be in the
wrong place over here.
443
00:24:10,390 --> 00:24:14,750
You see, again, the interesting
point is we don't
444
00:24:14,750 --> 00:24:17,800
have to rely on taking
my word for it.
445
00:24:17,800 --> 00:24:21,220
Somebody might say to me, now
look, all you've told me is
446
00:24:21,220 --> 00:24:25,120
that I get the wrong answer
solving this problem this
447
00:24:25,120 --> 00:24:26,600
particular way.
448
00:24:26,600 --> 00:24:29,040
And you've given me a nice
lecture about how the 2's come
449
00:24:29,040 --> 00:24:30,410
out the wrong way
and everything.
450
00:24:30,410 --> 00:24:33,456
How do I know that this
is the wrong answer?
451
00:24:33,456 --> 00:24:35,930
See, and again, everything
comes back to
452
00:24:35,930 --> 00:24:37,360
fundamentals again.
453
00:24:37,360 --> 00:24:42,500
To find 'd2y dx squared',
observe that by definition,
454
00:24:42,500 --> 00:24:46,210
that's just 'd dx' of 'dy dx'.
455
00:24:46,210 --> 00:24:49,220
456
00:24:49,220 --> 00:24:51,470
That definition doesn't depend
on what functions we're
457
00:24:51,470 --> 00:24:51,940
dealing with.
458
00:24:51,940 --> 00:24:54,300
The second derivative with
respect to 'x' is the
459
00:24:54,300 --> 00:24:57,450
derivative with respect to 'x'
of the first derivative.
460
00:24:57,450 --> 00:25:00,730
Now, once we have this, you see,
knowing from our previous
461
00:25:00,730 --> 00:25:03,620
case, that what?
462
00:25:03,620 --> 00:25:10,550
'dy dx' was 'f prime of t'
divided by 'g prime of t'.
463
00:25:10,550 --> 00:25:12,580
We can now do what?
464
00:25:12,580 --> 00:25:13,890
Take this derivative.
465
00:25:13,890 --> 00:25:16,240
By the way, again, notice
how the chain
466
00:25:16,240 --> 00:25:18,330
rule comes up in practice.
467
00:25:18,330 --> 00:25:20,220
It's not always dictated
to us.
468
00:25:20,220 --> 00:25:24,000
If you look at the expression
inside the parentheses, what
469
00:25:24,000 --> 00:25:25,110
do we have?
470
00:25:25,110 --> 00:25:28,400
Inside the parentheses, we have
a function of 't' only.
471
00:25:28,400 --> 00:25:29,830
This is a function of 't'.
472
00:25:29,830 --> 00:25:32,780
We want to differentiate
it with respect to 'x'.
473
00:25:32,780 --> 00:25:35,880
The most natural variable to
differentiate a function of
474
00:25:35,880 --> 00:25:39,210
't' with respect to
is 't' itself.
475
00:25:39,210 --> 00:25:42,370
In other words, what would've
been nice is if this was the
476
00:25:42,370 --> 00:25:47,780
derivative of 'f prime of t'
over 'g prime of t', with
477
00:25:47,780 --> 00:25:49,970
respect to 't'.
478
00:25:49,970 --> 00:25:51,760
See, this would be
easier to handle.
479
00:25:51,760 --> 00:25:53,330
We would then use the quotient
rule, et cetera.
480
00:25:53,330 --> 00:25:55,930
You see, we can differentiate
a function of 't'
481
00:25:55,930 --> 00:25:57,380
with respect to 't'.
482
00:25:57,380 --> 00:25:59,380
The trouble is we have
the derivative
483
00:25:59,380 --> 00:26:00,570
with respect to 'x'.
484
00:26:00,570 --> 00:26:03,885
And if we just change this to
a 't', that's cheating.
485
00:26:03,885 --> 00:26:06,600
See, I mean, you pretend you
copy it wrong, because it's an
486
00:26:06,600 --> 00:26:08,610
easier problem to
solve that way.
487
00:26:08,610 --> 00:26:12,010
The beauty of the chain rule is
that it allows us to do the
488
00:26:12,010 --> 00:26:15,900
problem the easier way, and
to doctor up the resulting
489
00:26:15,900 --> 00:26:18,300
incorrect answer by
the right answer.
490
00:26:18,300 --> 00:26:22,320
Namely, you see what we wanted
to wind up with here is what,
491
00:26:22,320 --> 00:26:24,860
the derivative not with respect
to 't', but with
492
00:26:24,860 --> 00:26:26,810
respect to 'x'.
493
00:26:26,810 --> 00:26:30,730
And so, by using the chain
rule, you see we do what?
494
00:26:30,730 --> 00:26:34,410
We take the derivative with
respect to 't', multiply that
495
00:26:34,410 --> 00:26:35,520
by 'dt dx'--
496
00:26:35,520 --> 00:26:38,950
again, mechanically, almost
as if these canceled.
497
00:26:38,950 --> 00:26:42,320
But this is the way the chain
rule works, and now, you see,
498
00:26:42,320 --> 00:26:45,450
I can work this out by
the regular quotient
499
00:26:45,450 --> 00:26:46,890
rule, which says what?
500
00:26:46,890 --> 00:26:53,460
It's the denominator times the
derivative of the numerator.
501
00:26:53,460 --> 00:26:55,530
See, and I am differentiating
out respect to 't', the
502
00:26:55,530 --> 00:27:02,530
natural variable, minus the
numerator times the derivative
503
00:27:02,530 --> 00:27:08,695
of the denominator over the
square of the denominator.
504
00:27:08,695 --> 00:27:12,790
505
00:27:12,790 --> 00:27:16,220
Now, that's a mess by itself,
meaning, what,
506
00:27:16,220 --> 00:27:17,930
computationally, it's
not that obvious.
507
00:27:17,930 --> 00:27:20,280
I mean, it's quite a bit of work
to do here, and then that
508
00:27:20,280 --> 00:27:27,370
whole thing must be multiplied
by 'dt dx'.
509
00:27:27,370 --> 00:27:31,500
And this, you see, is how one
goes around finding the second
510
00:27:31,500 --> 00:27:34,010
derivative of 'y' with respect
to 'x' in terms
511
00:27:34,010 --> 00:27:35,620
of parametric equations.
512
00:27:35,620 --> 00:27:38,630
And more than once, if you're
not careful, you're going to
513
00:27:38,630 --> 00:27:42,150
find yourself making serious
mistakes, by forgetting to put
514
00:27:42,150 --> 00:27:44,740
in this factor of 'dt dx'.
515
00:27:44,740 --> 00:27:48,450
By the way, an interesting
point is that we have not
516
00:27:48,450 --> 00:27:49,910
computed 'dt dx'.
517
00:27:49,910 --> 00:27:52,325
We have computed 'dx dt'.
518
00:27:52,325 --> 00:27:55,220
519
00:27:55,220 --> 00:27:56,680
Let's go back here.
520
00:27:56,680 --> 00:27:59,310
See, 'x' was 'g of t'.
521
00:27:59,310 --> 00:28:06,210
So from that, 'dx dt'
is 'g prime of t'.
522
00:28:06,210 --> 00:28:09,730
And the question is if 'dx dt'
is 'g prime of t', how does
523
00:28:09,730 --> 00:28:12,170
one find 'dt dx'?
524
00:28:12,170 --> 00:28:15,140
And again, I think your
intuition is going to tell you
525
00:28:15,140 --> 00:28:17,490
to just take reciprocals.
526
00:28:17,490 --> 00:28:21,230
And again, the question is it's
true that this suggests
527
00:28:21,230 --> 00:28:24,880
taking reciprocals, but how do
we know that we can do this,
528
00:28:24,880 --> 00:28:27,950
and if we can do this, what
does it really mean?
529
00:28:27,950 --> 00:28:30,730
You see, what this is leading
into is what's going to be the
530
00:28:30,730 --> 00:28:34,720
subject of our lecture next
time, called 'Inverse
531
00:28:34,720 --> 00:28:35,820
Functions'.
532
00:28:35,820 --> 00:28:38,550
And just to give you a preview
of what that lecture is about,
533
00:28:38,550 --> 00:28:41,450
and how we work things like
this, let's take a look at
534
00:28:41,450 --> 00:28:43,790
what we mean by inverse
functions.
535
00:28:43,790 --> 00:28:46,300
Well, we won't even mention
it in much detail.
536
00:28:46,300 --> 00:28:49,770
But let's take a look and see
what's going on over here.
537
00:28:49,770 --> 00:28:51,360
Let's suppose that the first--
538
00:28:51,360 --> 00:28:55,320
and by the way, I've started to
abandon using the 't' over
539
00:28:55,320 --> 00:28:56,450
here all the time.
540
00:28:56,450 --> 00:28:59,310
I think those of us in
engineering work primarily
541
00:28:59,310 --> 00:29:02,500
keep thinking of 't' as being
time, and you may get the
542
00:29:02,500 --> 00:29:06,940
mistaken notion that if the
variable isn't time, the thing
543
00:29:06,940 --> 00:29:09,000
doesn't work this way.
544
00:29:09,000 --> 00:29:11,870
In most cases, physically, the
variable that we're interested
545
00:29:11,870 --> 00:29:13,030
in will be time.
546
00:29:13,030 --> 00:29:15,780
But just for the idea of getting
you used to the fact
547
00:29:15,780 --> 00:29:17,780
that it makes no difference what
the name of the variable
548
00:29:17,780 --> 00:29:20,320
is, I've taken the liberty
of writing this slightly
549
00:29:20,320 --> 00:29:21,290
differently.
550
00:29:21,290 --> 00:29:24,570
Namely, I now assume that y is
a differentiable function of
551
00:29:24,570 --> 00:29:26,200
'u', and that 'u' is a
552
00:29:26,200 --> 00:29:28,130
differentiable function of 'x'.
553
00:29:28,130 --> 00:29:31,940
By the chain rule, I now know
that 'y' is a differentiable
554
00:29:31,940 --> 00:29:37,580
function of 'x', and that 'dy
dx' is 'dy du' times 'du dx'.
555
00:29:37,580 --> 00:29:40,990
The interesting thing here is,
is that there is nothing in
556
00:29:40,990 --> 00:29:44,080
the statement of the chain rule
that says that the first
557
00:29:44,080 --> 00:29:47,030
variable in the third that
'x' and 'y' must
558
00:29:47,030 --> 00:29:48,280
be different variables.
559
00:29:48,280 --> 00:29:51,820
In fact, it might happen that
'x' and 'y' are synonyms for
560
00:29:51,820 --> 00:29:53,100
one another.
561
00:29:53,100 --> 00:29:55,500
If 'x' and 'y' happen
to be synonyms--
562
00:29:55,500 --> 00:29:57,600
suppose 'x' and 'y'
are synonyms--
563
00:29:57,600 --> 00:29:59,460
look what happens over here.
564
00:29:59,460 --> 00:30:04,740
'dy dx' is then just 'dy
dy', which is 1.
565
00:30:04,740 --> 00:30:05,480
See, let's write that down.
566
00:30:05,480 --> 00:30:07,140
That's 'dy dy'.
567
00:30:07,140 --> 00:30:10,410
This would be 'dy du', and
if 'x' is equal to
568
00:30:10,410 --> 00:30:13,040
'y', this is 'du dy'.
569
00:30:13,040 --> 00:30:18,700
And if this is equal to 1, and
this is 'dy du', and this is
570
00:30:18,700 --> 00:30:23,160
'du dy', what does this tell
us about the relationship
571
00:30:23,160 --> 00:30:26,110
between 'dy du' and 'du dy'?
572
00:30:26,110 --> 00:30:28,840
It says their product is 1.
573
00:30:28,840 --> 00:30:32,250
And if the product is 1, that
by definition means that the
574
00:30:32,250 --> 00:30:35,330
two factors are reciprocals.
575
00:30:35,330 --> 00:30:38,440
Now, what I want you to observe
over here is what this
576
00:30:38,440 --> 00:30:39,990
whole thing means.
577
00:30:39,990 --> 00:30:44,450
Namely, if 'y' happens to equal
'x', do you see what
578
00:30:44,450 --> 00:30:45,610
this thing says?
579
00:30:45,610 --> 00:30:48,090
It says that 'y' is a
differentiable function of
580
00:30:48,090 --> 00:30:50,740
'u', and 'u' in turn is a
581
00:30:50,740 --> 00:30:53,050
differentiable function of 'y'.
582
00:30:53,050 --> 00:30:55,880
That's precisely what we
meant when we talked
583
00:30:55,880 --> 00:30:57,480
about inverse functions.
584
00:30:57,480 --> 00:31:00,440
We don't know when an inverse
function exists.
585
00:31:00,440 --> 00:31:03,960
All we're saying is, is that
if 'f inverse' happens to
586
00:31:03,960 --> 00:31:09,480
exist over here, to find 'du
dy', all we have to do is take
587
00:31:09,480 --> 00:31:12,560
the reciprocal of 'dy du'.
588
00:31:12,560 --> 00:31:15,240
Now again, this is
going to be the
589
00:31:15,240 --> 00:31:17,230
subject of our next lecture.
590
00:31:17,230 --> 00:31:20,550
All I wanted to do was to make
this aside for the time being.
591
00:31:20,550 --> 00:31:23,470
What I want to do to complete
today's lecture is to get to
592
00:31:23,470 --> 00:31:24,910
something more tangible.
593
00:31:24,910 --> 00:31:27,140
See, now that we've talked about
the chain rule, we've
594
00:31:27,140 --> 00:31:30,390
talked about inverse functions a
little bit, and talked about
595
00:31:30,390 --> 00:31:33,100
these things from a highly
theoretical point of view,
596
00:31:33,100 --> 00:31:34,980
let's go ahead and
try to solve a
597
00:31:34,980 --> 00:31:36,550
particularly simple problem.
598
00:31:36,550 --> 00:31:38,830
By particularly simple,
I mean this.
599
00:31:38,830 --> 00:31:42,230
I have chosen the numbers to
come out in a very, very easy
600
00:31:42,230 --> 00:31:45,580
way, so we don't get lost
in the maze of details.
601
00:31:45,580 --> 00:31:48,110
In other words, there was a
danger that we will confuse
602
00:31:48,110 --> 00:31:51,450
the computational details
with the theory.
603
00:31:51,450 --> 00:31:53,560
So to emphasize the theory,
I've tried to pick a
604
00:31:53,560 --> 00:31:55,820
straightforward simple problem,
but let's see how
605
00:31:55,820 --> 00:31:57,770
this thing works out.
606
00:31:57,770 --> 00:32:01,570
Let's suppose that we're given
that 'y' is equal to 't to the
607
00:32:01,570 --> 00:32:06,010
fourth power', and 'x' is
equal to 't squared'.
608
00:32:06,010 --> 00:32:08,850
What we would like to do-- and
by the way, notice what this
609
00:32:08,850 --> 00:32:13,050
thing says, a given value of 't'
determines both an 'x' and
610
00:32:13,050 --> 00:32:17,530
the 'y', so that makes 'x' and
'y' functionally related.
611
00:32:17,530 --> 00:32:21,200
Notice that from the first
equation, we can find that 'dy
612
00:32:21,200 --> 00:32:25,350
dt' is '4 t cubed'.
613
00:32:25,350 --> 00:32:30,900
From the second equation, we can
find that 'dx dt' is '2t'.
614
00:32:30,900 --> 00:32:36,280
And if we now use the chain
rule, 'dy dx' will be what?
615
00:32:36,280 --> 00:32:42,580
It'll be 'dy dt' divided
by 'dx dt', and
616
00:32:42,580 --> 00:32:48,140
that's just '2 t squared'.
617
00:32:48,140 --> 00:32:52,890
By the way, as a check,
notice this.
618
00:32:52,890 --> 00:32:56,150
If 'y' is equal to 't to the
fourth', and 'x' is equal to
619
00:32:56,150 --> 00:32:59,970
't squared', since 't to the
fourth' is the square of 't
620
00:32:59,970 --> 00:33:04,280
squared', that says 'y' is
equal to ''t squared'
621
00:33:04,280 --> 00:33:08,900
squared', 'y' is equal
to 'x squared'.
622
00:33:08,900 --> 00:33:12,180
And if 'y' is equal to 'x
squared', in this case, it's
623
00:33:12,180 --> 00:33:16,800
very easy to see that 'dy
dx' is equal to '2x'.
624
00:33:16,800 --> 00:33:20,530
By the way, when we try to
compare these two answers,
625
00:33:20,530 --> 00:33:22,340
they look different, but
that's because they're
626
00:33:22,340 --> 00:33:24,750
expressed in terms of
different variables.
627
00:33:24,750 --> 00:33:30,430
If we return to our original
equations, and we see that 'x'
628
00:33:30,430 --> 00:33:32,530
is equal to 't squared'--
629
00:33:32,530 --> 00:33:34,470
'x' is a synonym for
't squared'--
630
00:33:34,470 --> 00:33:38,080
this is the check that we have
received the right answer.
631
00:33:38,080 --> 00:33:41,400
By the way, before I conclude
today's lecture, I would like
632
00:33:41,400 --> 00:33:43,630
to make a rather
important aside
633
00:33:43,630 --> 00:33:46,060
about parametric equations.
634
00:33:46,060 --> 00:33:49,500
After one works the problem this
way, and comes down to
635
00:33:49,500 --> 00:33:53,300
the check, and says, hey, after
all of this mess over
636
00:33:53,300 --> 00:33:56,610
here, I could have replaced
it by just 'y' equals 'x
637
00:33:56,610 --> 00:33:59,000
squared', why did I have
to work with this
638
00:33:59,000 --> 00:34:00,220
in the first place?
639
00:34:00,220 --> 00:34:02,770
We are going to have many, many
examples throughout the
640
00:34:02,770 --> 00:34:04,770
course that will illustrate
this.
641
00:34:04,770 --> 00:34:07,440
But at least once in a lecture,
I would like to go on
642
00:34:07,440 --> 00:34:12,380
record as pointing out that this
pair of equations tells
643
00:34:12,380 --> 00:34:15,250
you much more than this
equation here.
644
00:34:15,250 --> 00:34:17,909
This equation simply
tells you this.
645
00:34:17,909 --> 00:34:21,880
If a particle were moving along
a curve with respect to
646
00:34:21,880 --> 00:34:26,139
time according to these
equations, this equation here
647
00:34:26,139 --> 00:34:29,050
simply tells you what path the
particle would follow.
648
00:34:29,050 --> 00:34:34,480
Namely, the parabola 'y'
equals 'x squared'.
649
00:34:34,480 --> 00:34:38,100
On the other hand, these
two equations tell you
650
00:34:38,100 --> 00:34:39,420
much more than that.
651
00:34:39,420 --> 00:34:43,719
These not only tell you that the
particle moved along the
652
00:34:43,719 --> 00:34:47,310
parabola 'y' equals 'x squared',
but rather, it tells
653
00:34:47,310 --> 00:34:53,020
you at a particular time the
point on the parabola that the
654
00:34:53,020 --> 00:34:54,630
particle was at.
655
00:34:54,630 --> 00:34:56,130
What I mean is this.
656
00:34:56,130 --> 00:35:00,480
As another example, suppose we
had 'y' equals 't squared,'
657
00:35:00,480 --> 00:35:02,920
and 'x' equals 't'.
658
00:35:02,920 --> 00:35:07,090
If we eliminate 't' from these
two equations, we also find
659
00:35:07,090 --> 00:35:09,960
that 'y' is equal
to 'x squared'.
660
00:35:09,960 --> 00:35:15,050
Yet notice that this is not the
same as our original set
661
00:35:15,050 --> 00:35:16,040
of equations.
662
00:35:16,040 --> 00:35:20,340
For example, here, when 't' is
2, when 't' is 2 over here,
663
00:35:20,340 --> 00:35:24,260
what point are we on as far as
the parabola is concerned?
664
00:35:24,260 --> 00:35:27,930
When 't' is 2, this is
2, and this is 4.
665
00:35:27,930 --> 00:35:30,760
That would be the
point 2 comma 4.
666
00:35:30,760 --> 00:35:33,960
On the other hand, with respect
to this equation, when
667
00:35:33,960 --> 00:35:42,230
't' is 2, 'x' is 4, and 'y' is
16, you see both of these
668
00:35:42,230 --> 00:35:47,940
particles would follow the same
curve, but they are at
669
00:35:47,940 --> 00:35:50,500
different points at
different times.
670
00:35:50,500 --> 00:35:53,360
So don't belittle the
parametric approach.
671
00:35:53,360 --> 00:35:56,980
Having the parameter 't' in
there tells you more than just
672
00:35:56,980 --> 00:35:59,680
what the path of
the motion is.
673
00:35:59,680 --> 00:36:02,380
It tells you at what time
a particle was at what
674
00:36:02,380 --> 00:36:03,700
particular point.
675
00:36:03,700 --> 00:36:05,550
Well, enough about that.
676
00:36:05,550 --> 00:36:08,870
Let's go ahead and find the
second derivative now.
677
00:36:08,870 --> 00:36:12,770
You see, we already know that
'dy dx' is '2 t squared'.
678
00:36:12,770 --> 00:36:18,230
Now what we'd like to find
is 'd2y dx squared'.
679
00:36:18,230 --> 00:36:20,710
Again, the same basic
definition.
680
00:36:20,710 --> 00:36:22,260
'd2y dx squared'.
681
00:36:22,260 --> 00:36:24,930
The second derivative is the
derivative of the first
682
00:36:24,930 --> 00:36:26,970
derivative.
683
00:36:26,970 --> 00:36:30,600
The first derivative we saw was
'2 t squared', so this is
684
00:36:30,600 --> 00:36:33,200
the derivative of
'2 t squared'.
685
00:36:33,200 --> 00:36:35,470
Again, and this is where most
of the mistakes are made,
686
00:36:35,470 --> 00:36:37,000
people get sloppy.
687
00:36:37,000 --> 00:36:38,720
They forget the 'x'
is in here.
688
00:36:38,720 --> 00:36:41,710
They say I know the derivative
of this, it's '4t'.
689
00:36:41,710 --> 00:36:45,600
Well, the derivative of this is
'4t' with respect to 't'.
690
00:36:45,600 --> 00:36:49,560
We want to differentiate
with respect to 'x'.
691
00:36:49,560 --> 00:36:54,020
And the way the chain rule comes
in, we say OK, since 't'
692
00:36:54,020 --> 00:36:56,660
is the natural variable with
respect to which to
693
00:36:56,660 --> 00:36:58,470
differentiate, let's do it.
694
00:36:58,470 --> 00:37:00,820
We'll differentiate
in respect to 't'.
695
00:37:00,820 --> 00:37:04,270
But since the final answer has
to be with respect to 'x', our
696
00:37:04,270 --> 00:37:08,590
correction factor by the chain
rule will be 'dt dx'.
697
00:37:08,590 --> 00:37:11,710
Well, the derivative of '2 t
squared' with respect to 't'
698
00:37:11,710 --> 00:37:13,750
is clearly '4t'.
699
00:37:13,750 --> 00:37:16,860
The derivative of 't' with
respect to 'x', assuming that
700
00:37:16,860 --> 00:37:19,470
we know something about inverse
functions, that's the
701
00:37:19,470 --> 00:37:21,680
reciprocal of 'dx dt'.
702
00:37:21,680 --> 00:37:25,270
We just saw that 'dx
dt' was '2t'.
703
00:37:25,270 --> 00:37:28,950
Therefore, 'dt dx' is 1 over
'2t', and therefore, the
704
00:37:28,950 --> 00:37:31,870
correct answer appears
to be 2.
705
00:37:31,870 --> 00:37:35,330
Again, this is why I picked
the simple case.
706
00:37:35,330 --> 00:37:38,800
Given that 'y' equals 'x
squared', we see at a glance
707
00:37:38,800 --> 00:37:42,520
that 'dy dx' is equal to '2x',
and also at a glance,
708
00:37:42,520 --> 00:37:48,330
therefore, that 'd2y dx squared'
is equal to 2.
709
00:37:48,330 --> 00:37:51,520
By the way, that's exactly
what this is equal to.
710
00:37:51,520 --> 00:37:54,660
You see, had we forgot the chain
rule, and had we left
711
00:37:54,660 --> 00:37:57,610
this factor out, this would
have given us--
712
00:37:57,610 --> 00:37:59,530
in other words, to simply write
down that the answer was
713
00:37:59,530 --> 00:38:02,510
'4t', which is the most common
mistake that's made, would
714
00:38:02,510 --> 00:38:04,300
have given us the
wrong answer.
715
00:38:04,300 --> 00:38:06,300
That's why I put such
an easy problem.
716
00:38:06,300 --> 00:38:09,150
You see, if I had picked a
tougher computational problem,
717
00:38:09,150 --> 00:38:10,940
the theory would have
remained the same.
718
00:38:10,940 --> 00:38:13,190
But when I got two different
answers, it would have been
719
00:38:13,190 --> 00:38:17,380
difficult to determine which
was the correct answer, and
720
00:38:17,380 --> 00:38:19,420
which was the incorrect
answer.
721
00:38:19,420 --> 00:38:22,140
But again, to summarize today's
lecture, it was a
722
00:38:22,140 --> 00:38:25,510
continuation in a way of the
lecture of last time, when we
723
00:38:25,510 --> 00:38:29,370
developed the primary recipe
involving differentials.
724
00:38:29,370 --> 00:38:31,440
Now we applied that
to find something
725
00:38:31,440 --> 00:38:33,190
called the chain rule.
726
00:38:33,190 --> 00:38:37,130
In the process of emphasizing
the chain rule, we talked
727
00:38:37,130 --> 00:38:39,320
about the necessity of
knowing something
728
00:38:39,320 --> 00:38:41,200
about inverse functions.
729
00:38:41,200 --> 00:38:45,440
Consequently, that dictates what
our next lecture will be
730
00:38:45,440 --> 00:38:49,100
concerned with, namely
inverse functions.
731
00:38:49,100 --> 00:38:51,390
And so until next
time, goodbye.
732
00:38:51,390 --> 00:38:54,730
733
00:38:54,730 --> 00:38:57,270
ANNOUNCER: Funding for the
publication of this video was
734
00:38:57,270 --> 00:39:01,980
provided by the Gabriella and
Paul Rosenbaum Foundation.
735
00:39:01,980 --> 00:39:06,160
Help OCW continue to provide
free and open access to MIT
736
00:39:06,160 --> 00:39:10,350
courses by making a donation
at ocw.mit.edu/donate.
737
00:39:10,350 --> 00:39:15,092