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PROFESSOR: Right now,
we're finishing up
00:00:25.000 --> 00:00:27.740
with the first
unit, and I'd like
00:00:27.740 --> 00:00:31.890
to continue in this
lecture, lecture seven,
00:00:31.890 --> 00:00:45.510
with some final remarks
about exponents.
00:00:45.510 --> 00:00:49.560
So what I'd like to do
is just review something
00:00:49.560 --> 00:00:51.910
that I did quickly
last time, and make
00:00:51.910 --> 00:00:53.920
a few philosophical
remarks about it.
00:00:53.920 --> 00:00:57.660
I think that the steps involved
were maybe a little tricky,
00:00:57.660 --> 00:01:00.680
and so I'd like to go
through it one more time.
00:01:00.680 --> 00:01:03.900
Remember, that we were
talking about this number a_k,
00:01:03.900 --> 00:01:05.510
which is (1 + 1/k)^k.
00:01:08.270 --> 00:01:11.400
And what we showed was
that the limit as k
00:01:11.400 --> 00:01:16.920
goes to infinity of a_k was e.
00:01:16.920 --> 00:01:19.590
So the first thing
that I'd like to do
00:01:19.590 --> 00:01:22.890
is just explain the proof
a little bit more clearly
00:01:22.890 --> 00:01:28.440
than I did last time
with fewer symbols,
00:01:28.440 --> 00:01:31.830
or at least with this
abbreviation of the symbol
00:01:31.830 --> 00:01:35.050
here, to show you
what we actually did.
00:01:35.050 --> 00:01:41.580
So I'll just remind you
of what we did last time,
00:01:41.580 --> 00:01:45.726
and the first
observation was to check,
00:01:45.726 --> 00:01:47.350
rather than the limit
of this function,
00:01:47.350 --> 00:01:49.550
but to take the log first.
00:01:49.550 --> 00:01:51.640
And this is
typically what's done
00:01:51.640 --> 00:01:54.810
when you have an exponential,
when you have an exponent.
00:01:54.810 --> 00:01:57.750
And what we found was
that the limit here
00:01:57.750 --> 00:02:03.500
was 1 as k goes to infinity.
00:02:03.500 --> 00:02:05.160
So last time, this
is what we did.
00:02:05.160 --> 00:02:07.970
And I just wanted to
be careful and show you
00:02:07.970 --> 00:02:09.830
exactly what the next step is.
00:02:09.830 --> 00:02:14.360
If you exponentiate this fact;
you take e to this power,
00:02:14.360 --> 00:02:21.230
that's going to tend to
e^1, which is just e.
00:02:21.230 --> 00:02:26.570
And then, we just observe
that this is the same as a_k.
00:02:26.570 --> 00:02:32.180
So the basic ingredient
here is that e^ln a = a.
00:02:32.180 --> 00:02:36.750
That's because the log
function is the inverse
00:02:36.750 --> 00:02:38.030
of the exponential function.
00:02:38.030 --> 00:02:38.860
Yes, question?
00:02:38.860 --> 00:02:54.190
STUDENT: [INAUDIBLE]
00:02:54.190 --> 00:02:58.180
PROFESSOR: So the question
was, wouldn't the log of this
00:02:58.180 --> 00:03:01.530
be 0 because a_k
is tending to 1.
00:03:01.530 --> 00:03:03.830
But a_k isn't tending to 1.
00:03:03.830 --> 00:03:06.640
Who said it was?
00:03:06.640 --> 00:03:10.040
If you take the logarithm,
which is what we did last time,
00:03:10.040 --> 00:03:12.880
logarithm of a_k is
indeed k * ln(1 + 1/k).
00:03:17.030 --> 00:03:18.520
That does not tend to 0.
00:03:18.520 --> 00:03:22.610
This part of it tends to 0, and
this part tends to infinity.
00:03:22.610 --> 00:03:26.460
And they balance each
other, 0 times infinity.
00:03:26.460 --> 00:03:28.460
We don't really know yet
from this expression,
00:03:28.460 --> 00:03:32.189
in fact we did some cleverness
with limits and derivatives,
00:03:32.189 --> 00:03:33.230
to figure out this limit.
00:03:33.230 --> 00:03:34.354
It was a very subtle thing.
00:03:34.354 --> 00:03:37.560
It turned out to be 1.
00:03:37.560 --> 00:03:38.850
All right?
00:03:38.850 --> 00:03:40.645
Now, the thing that
I'd like to say
00:03:40.645 --> 00:03:43.550
- I'm sorry I'm going
to erase this aside here
00:03:43.550 --> 00:03:46.100
- but you need to go
back to your notes
00:03:46.100 --> 00:03:48.824
and remember that this
is what we did last time.
00:03:48.824 --> 00:03:50.490
Because I want to
have room for the next
00:03:50.490 --> 00:03:54.420
comment that I want to make on
this little blackboard here.
00:03:54.420 --> 00:03:57.090
What we just derived
was this property here,
00:03:57.090 --> 00:04:01.810
but I made a claim
yesterday, and I just
00:04:01.810 --> 00:04:04.120
want to emphasize it
again so that we realized
00:04:04.120 --> 00:04:07.786
what it is that we're doing.
00:04:07.786 --> 00:04:08.910
I looked at this backwards.
00:04:08.910 --> 00:04:11.430
One way you can think of this
is we're evaluating this limit
00:04:11.430 --> 00:04:13.000
and getting an answer.
00:04:13.000 --> 00:04:16.524
But all equalities can
be read both directions.
00:04:16.524 --> 00:04:17.940
And we can write
it the other way:
00:04:17.940 --> 00:04:25.881
e equals the limit, as k goes
to infinity, of this expression
00:04:25.881 --> 00:04:26.380
here.
00:04:26.380 --> 00:04:28.640
So that's just the same thing.
00:04:28.640 --> 00:04:30.840
And if we read it
backwards, what we're saying
00:04:30.840 --> 00:04:35.700
is that this limit
is a formula for e.
00:04:35.700 --> 00:04:38.325
So this is very
typical of mathematics.
00:04:38.325 --> 00:04:40.700
You want to always reverse
your perspective all the time.
00:04:40.700 --> 00:04:43.710
Equations work both
ways, and in this case,
00:04:43.710 --> 00:04:46.300
we have two different
things here.
00:04:46.300 --> 00:04:49.430
This e was what we
defined as the base,
00:04:49.430 --> 00:04:54.461
which when you graph e^x,
you get slope 1 at 0.
00:04:54.461 --> 00:04:56.710
And then it turns out to be
equal to this limit, which
00:04:56.710 --> 00:04:59.060
we can calculate numerically.
00:04:59.060 --> 00:05:02.100
If you do this on your
calculators, you, of course,
00:05:02.100 --> 00:05:05.420
will have a way of
programming in this number
00:05:05.420 --> 00:05:07.290
and evaluating it for each k.
00:05:07.290 --> 00:05:10.880
And you'll have another button
available to evaluate this one.
00:05:10.880 --> 00:05:12.432
So another way of
saying it is it
00:05:12.432 --> 00:05:14.640
that there's a relationship
between these two things.
00:05:14.640 --> 00:05:19.210
And all of calculus is a matter
of getting these relationships.
00:05:19.210 --> 00:05:21.860
So we can look at these things
in several different ways.
00:05:21.860 --> 00:05:23.350
And indeed, that's
what we're going
00:05:23.350 --> 00:05:25.832
to be doing at least
at the end of today
00:05:25.832 --> 00:05:27.040
in talking about derivatives.
00:05:27.040 --> 00:05:29.480
A lot of times when we
talk about derivatives,
00:05:29.480 --> 00:05:32.230
we're trying to look at them
from several perspectives
00:05:32.230 --> 00:05:34.780
at once.
00:05:34.780 --> 00:05:37.280
Okay, so I have to keep
on going with exponents,
00:05:37.280 --> 00:05:40.130
because I have one loose end.
00:05:40.130 --> 00:05:44.490
One loose end that
I did not cover yet.
00:05:44.490 --> 00:05:48.330
There's one very important
formula that's left,
00:05:48.330 --> 00:05:51.920
and it's the derivative
of the powers.
00:05:51.920 --> 00:05:54.070
We actually didn't
do this - well we
00:05:54.070 --> 00:05:57.120
did it for rational numbers r.
00:05:57.120 --> 00:06:00.070
So this is the formula here.
00:06:00.070 --> 00:06:06.200
But now we're going to check
this for all real numbers, r.
00:06:06.200 --> 00:06:09.200
So including all the
irrational ones as well.
00:06:09.200 --> 00:06:13.690
This is also good
practice for using base e
00:06:13.690 --> 00:06:16.480
and using logarithmic
differentiation.
00:06:16.480 --> 00:06:20.810
So let me do this
by our two methods
00:06:20.810 --> 00:06:26.160
that we can use to handle
exponential type problems.
00:06:26.160 --> 00:06:32.070
So method one was base e.
00:06:32.070 --> 00:06:34.040
So if I just rewrite
this base e again,
00:06:34.040 --> 00:06:36.910
that's just this
formula over here.
00:06:36.910 --> 00:06:50.350
x^r = (e^ln x)^r,
which is e^r ln x.
00:06:50.350 --> 00:06:55.320
Okay, so now I can
differentiate this.
00:06:55.320 --> 00:07:04.650
So I get that d/dx (x^r),
now I'm going to use prime
00:07:04.650 --> 00:07:07.460
notation, because I don't want
to keep on writing that d/dx
00:07:07.460 --> 00:07:10.580
here; (e^(r ln x))'.
00:07:13.890 --> 00:07:18.457
And now, what I can do is
I can use the chain rule.
00:07:18.457 --> 00:07:20.290
The chain rule says
that it's the derivative
00:07:20.290 --> 00:07:24.380
of this times the
derivative of the function.
00:07:24.380 --> 00:07:29.500
So the derivative of the
exponential is just itself.
00:07:29.500 --> 00:07:31.470
And the derivative
of this guy here,
00:07:31.470 --> 00:07:35.400
well I'll write it out
once, is (r ln x)'.
00:07:39.700 --> 00:07:42.250
So what's that equal to?
00:07:42.250 --> 00:07:45.890
Well, e^(r ln x) is is just x^r.
00:07:45.890 --> 00:07:53.870
And this derivative here is--
Well the derivative of r is 0.
00:07:53.870 --> 00:07:54.860
This is a constant.
00:07:54.860 --> 00:07:56.680
It just factors out.
00:07:56.680 --> 00:08:02.070
And ln x now has derivative--
What's the derivative of ln x?
00:08:02.070 --> 00:08:06.570
1/x, so this is going
to be times r/x.
00:08:06.570 --> 00:08:10.180
And now, we rewrite it in the
customary form, which is r,
00:08:10.180 --> 00:08:13.040
we put the r in front, x^(r-1).
00:08:13.040 --> 00:08:13.820
Okay?
00:08:13.820 --> 00:08:19.060
So I just derived
the formula for you.
00:08:19.060 --> 00:08:23.310
And it didn't matter whether
r was rational or irrational,
00:08:23.310 --> 00:08:25.320
it's the same proof.
00:08:25.320 --> 00:08:29.440
Okay so now I have to show you
how method two works as well.
00:08:29.440 --> 00:08:34.550
So let's do method
two, which we call
00:08:34.550 --> 00:08:39.280
logarithmic differentiation.
00:08:39.280 --> 00:08:43.764
And so here I'll use a
symbol, say u, for x^r,
00:08:43.764 --> 00:08:44.930
and I'll take its logarithm.
00:08:44.930 --> 00:08:50.480
That's r ln x.
00:08:50.480 --> 00:08:51.830
And now I differentiate it.
00:08:51.830 --> 00:08:54.160
I'll leave that in
the middle, because I
00:08:54.160 --> 00:08:55.979
want to remember
the key property
00:08:55.979 --> 00:08:57.270
of logarithmic differentiation.
00:08:57.270 --> 00:08:58.747
But first I'll differentiate it.
00:08:58.747 --> 00:09:01.080
Later on, what I'm going to
use is that this is the same
00:09:01.080 --> 00:09:02.620
as u'/u.
00:09:02.620 --> 00:09:06.470
This is one way of evaluating
a logarithmic derivative.
00:09:06.470 --> 00:09:08.660
And then the other
is to differentiate
00:09:08.660 --> 00:09:10.980
the explicit function
that we have over here.
00:09:10.980 --> 00:09:16.790
And that is just,
as we said, r/x.
00:09:16.790 --> 00:09:25.470
So now, I multiply through, and
I get u' = ur/x which is just
00:09:25.470 --> 00:09:29.750
x^r r/x, which is just
what we did before.
00:09:29.750 --> 00:09:30.520
It's r x^(r-1).
00:09:33.410 --> 00:09:36.790
Again, you can now
see by comparing
00:09:36.790 --> 00:09:40.110
these two pieces of arithmetic
that they're basically
00:09:40.110 --> 00:09:41.470
the same.
00:09:41.470 --> 00:09:43.510
Pretty much every time
you convert to base e
00:09:43.510 --> 00:09:45.170
or you do logarithmic
differentiation,
00:09:45.170 --> 00:09:46.836
it'll amount to the
same thing, provided
00:09:46.836 --> 00:09:48.270
you don't get mixed up.
00:09:48.270 --> 00:09:51.720
You generally have to
introduce a new symbol here.
00:09:51.720 --> 00:09:55.840
On the other hand, you're
dealing with exponents there.
00:09:55.840 --> 00:10:00.990
It's worth it to know
both points of view.
00:10:00.990 --> 00:10:07.470
All right, so now I want to
make one last remark before we
00:10:07.470 --> 00:10:09.910
finish with exponents.
00:10:09.910 --> 00:10:16.120
And, I'll try to sell this
to you in a lot of ways
00:10:16.120 --> 00:10:19.490
as the course goes on,
but one thing that I
00:10:19.490 --> 00:10:23.370
want to try to emphasize is that
the natural logarithm really
00:10:23.370 --> 00:10:27.210
is natural.
00:10:27.210 --> 00:10:39.920
So, I claim that the
natural log is natural.
00:10:39.920 --> 00:10:45.900
And the example that we're going
to use for this illustration
00:10:45.900 --> 00:10:53.340
is economics.
00:10:53.340 --> 00:10:54.090
Okay?
00:10:54.090 --> 00:10:58.520
So let me explain to why the
natural log is the one that's
00:10:58.520 --> 00:11:00.820
natural for economics.
00:11:00.820 --> 00:11:06.040
If you are imagining the
price of a stock that you own
00:11:06.040 --> 00:11:11.160
goes down by a dollar, that's a
totally meaningless statement.
00:11:11.160 --> 00:11:13.439
It depends on a lot of things.
00:11:13.439 --> 00:11:15.730
In particular, it depends on
whether the original price
00:11:15.730 --> 00:11:18.300
was a dollar or 100 dollars.
00:11:18.300 --> 00:11:22.130
So there's not much meaning
to these absolute numbers.
00:11:22.130 --> 00:11:25.080
It's always the
ratios that matter.
00:11:25.080 --> 00:11:29.280
So, for example, I just
looked up an hour ago,
00:11:29.280 --> 00:11:42.050
the London Exchange closed,
and it was down 27.9,
00:11:42.050 --> 00:11:44.480
which as I said, is
pretty meaningless
00:11:44.480 --> 00:11:50.050
unless you know what the
actual total of this index is.
00:11:50.050 --> 00:11:54.200
It turns out it was 6,432.
00:11:54.200 --> 00:11:57.070
So the change in
the price, divided
00:11:57.070 --> 00:12:03.980
by the price, which in
this case is 27.9 / 6,432,
00:12:03.980 --> 00:12:07.550
is what matters.
00:12:07.550 --> 00:12:11.791
And, in this case, it
happens to be .43%.
00:12:11.791 --> 00:12:12.290
All right?
00:12:12.290 --> 00:12:14.270
That's what happened today.
00:12:14.270 --> 00:12:18.410
And similarly, if you take
the infinitesimal of this,
00:12:18.410 --> 00:12:21.260
people think of days as being
relatively small increments
00:12:21.260 --> 00:12:23.900
when you're
investing in a stock,
00:12:23.900 --> 00:12:27.240
you would be interested in
the infinitesimal sense,
00:12:27.240 --> 00:12:28.690
you would be interested in p'/p.
00:12:28.690 --> 00:12:33.080
The derivative of
p divided by p.
00:12:33.080 --> 00:12:35.530
That's just (ln p)'.
00:12:38.160 --> 00:12:42.275
So this is the - let me
put a little box around it
00:12:42.275 --> 00:12:45.460
- the formula of
logarithmic differentiation.
00:12:45.460 --> 00:12:49.700
But let me just emphasize that
it has an actual significance,
00:12:49.700 --> 00:12:52.430
and it's the one that's used
by economists and people who
00:12:52.430 --> 00:12:54.450
are modeling prices of
things all the time.
00:12:54.450 --> 00:12:58.620
They never use absolute prices
when there are large swings.
00:12:58.620 --> 00:13:01.010
They always use the
log of the price.
00:13:01.010 --> 00:13:07.010
And there's no point in using
log base 10, or log base 2.
00:13:07.010 --> 00:13:08.180
Those give you junk.
00:13:08.180 --> 00:13:11.190
They give you an
extra factor of log 2.
00:13:11.190 --> 00:13:14.870
It's the natural log that's
the obvious one to use.
00:13:14.870 --> 00:13:18.000
It's completely
straightforward that this
00:13:18.000 --> 00:13:21.010
is a simpler expression
than using log base 10
00:13:21.010 --> 00:13:24.030
and having a factor of
natural log of 10 there,
00:13:24.030 --> 00:13:26.800
which would just
mess everything up.
00:13:26.800 --> 00:13:29.360
All right, so this is
just one illustration.
00:13:29.360 --> 00:13:31.680
Anything that has
to do with ratios
00:13:31.680 --> 00:13:36.160
is going to
encounter logarithms.
00:13:36.160 --> 00:13:41.270
All right, so that's
pretty much it.
00:13:41.270 --> 00:13:45.822
That's all I want to
say for now anyway.
00:13:45.822 --> 00:13:47.280
There's lots more
to say, but we'll
00:13:47.280 --> 00:13:50.459
be saying it when we do
applications of derivatives
00:13:50.459 --> 00:13:51.250
in the second unit.
00:13:51.250 --> 00:13:54.450
So now, what I'd like to
do is to start a review.
00:13:54.450 --> 00:13:57.790
I'm just going to run through
what we did in this unit.
00:13:57.790 --> 00:13:59.700
I'll tell you
approximately what I
00:13:59.700 --> 00:14:06.150
expect from you on the test
that's coming up tomorrow.
00:14:06.150 --> 00:14:14.640
And, well, so let's
get started with that.
00:14:14.640 --> 00:14:27.050
So this is a review of Unit One.
00:14:27.050 --> 00:14:32.610
And I'm just going to put on
the board all of the things
00:14:32.610 --> 00:14:35.750
that you need to think about,
anyway, keep in your head.
00:14:35.750 --> 00:14:41.750
And there are what are
called general formulas
00:14:41.750 --> 00:14:45.070
for derivatives.
00:14:45.070 --> 00:14:51.970
And then there are
the specific ones.
00:14:51.970 --> 00:14:55.920
And let me just remind you
what the general formulas are.
00:14:55.920 --> 00:14:58.750
There's what you
do to differentiate
00:14:58.750 --> 00:15:04.190
a sum, a multiple of a
function, the product
00:15:04.190 --> 00:15:08.190
rule, the quotient rule.
00:15:08.190 --> 00:15:11.550
Those are several
general formulas.
00:15:11.550 --> 00:15:13.080
And then there's
one more, which is
00:15:13.080 --> 00:15:15.780
the chain rule, which
I'm going to say just
00:15:15.780 --> 00:15:17.450
a little bit more about.
00:15:17.450 --> 00:15:21.200
So the derivative of a
function of a function
00:15:21.200 --> 00:15:26.380
is the derivative of the
function times the derivative
00:15:26.380 --> 00:15:27.430
of the other function.
00:15:27.430 --> 00:15:33.780
So here, I've
abbreviated u is u(x).
00:15:33.780 --> 00:15:36.630
Right, so this is one of
two ways of writing it.
00:15:36.630 --> 00:15:39.850
The other way is also one
that you can keep in mind
00:15:39.850 --> 00:15:42.470
and you might find
easier to remember.
00:15:42.470 --> 00:15:46.690
It's probably a good idea
to remember both formulas.
00:15:46.690 --> 00:15:49.660
And then the last type
of general formula
00:15:49.660 --> 00:15:56.950
that we did was implicit
differentiation.
00:15:56.950 --> 00:15:59.200
Okay?
00:15:59.200 --> 00:16:03.190
So when you do implicit
differentiation,
00:16:03.190 --> 00:16:06.530
you have an equation
and you don't
00:16:06.530 --> 00:16:09.270
try to solve for the
unknown function.
00:16:09.270 --> 00:16:13.110
You just put it in its simplest
form and you differentiate.
00:16:13.110 --> 00:16:20.440
So, we actually did this,
in particular, for inverses.
00:16:20.440 --> 00:16:23.520
That was a very, very key
method for calculating
00:16:23.520 --> 00:16:25.180
the inverses of functions.
00:16:25.180 --> 00:16:28.600
And it's also true that
logarithmic differentiation
00:16:28.600 --> 00:16:31.420
is of this type.
00:16:31.420 --> 00:16:33.299
This is a transformation.
00:16:33.299 --> 00:16:34.840
We're differentiating
something else.
00:16:34.840 --> 00:16:37.920
We're transforming the equation
by taking its logarithm
00:16:37.920 --> 00:16:40.980
and then differentiating.
00:16:40.980 --> 00:16:45.200
Okay, so there are a number of
different ways this is applied.
00:16:45.200 --> 00:16:48.450
It can also be applied,
anyway, these are two of them.
00:16:48.450 --> 00:16:50.320
So maybe in parenthesis.
00:16:50.320 --> 00:16:53.120
These are just examples.
00:16:53.120 --> 00:16:54.350
All right.
00:16:54.350 --> 00:16:59.461
I'll try to give examples of
at least a few of these rules
00:16:59.461 --> 00:16:59.960
later.
00:16:59.960 --> 00:17:05.670
So now, the specific
functions that you know how
00:17:05.670 --> 00:17:08.360
to differentiate: well you know
how to differentiate now x^r
00:17:08.360 --> 00:17:11.410
thanks to what I just did.
00:17:11.410 --> 00:17:15.030
We have the sine and
the cosine functions,
00:17:15.030 --> 00:17:17.910
which you're
responsible for knowing
00:17:17.910 --> 00:17:19.500
what their derivatives are.
00:17:19.500 --> 00:17:26.490
And then other trig functions
like tan and secant.
00:17:26.490 --> 00:17:29.810
We generally don't bother
with cosecants and cotangents,
00:17:29.810 --> 00:17:32.710
because everything can be
expressed in terms of these
00:17:32.710 --> 00:17:33.759
anyway.
00:17:33.759 --> 00:17:35.550
Actually, you can really
express everything
00:17:35.550 --> 00:17:36.906
in terms of sines and cosines.
00:17:36.906 --> 00:17:38.280
But what you'll
find is that it's
00:17:38.280 --> 00:17:41.660
much more convenient to remember
the derivatives of these
00:17:41.660 --> 00:17:42.730
as well.
00:17:42.730 --> 00:17:45.870
So memorize all of these.
00:17:45.870 --> 00:17:49.810
All right, and then
we had e^x and ln x.
00:17:49.810 --> 00:17:53.920
And we had the inverses
of the trig functions.
00:17:53.920 --> 00:18:00.010
These were the two that we did:
the arctangent and the arcsine.
00:18:00.010 --> 00:18:02.220
So those are the ones
you're responsible for.
00:18:02.220 --> 00:18:06.970
You should have enough time,
anyway, to work out anything
00:18:06.970 --> 00:18:09.390
else, if you know these.
00:18:09.390 --> 00:18:11.210
All right, so
basically the idea is
00:18:11.210 --> 00:18:13.070
you have a bunch of
special formulas.
00:18:13.070 --> 00:18:14.820
You have a bunch of
general formulas.
00:18:14.820 --> 00:18:16.620
You put them
together, and you can
00:18:16.620 --> 00:18:20.970
generate pretty much anything.
00:18:20.970 --> 00:18:24.810
Okay, so let's do a few
examples before going on
00:18:24.810 --> 00:18:41.290
with the review.
00:18:41.290 --> 00:18:48.230
Okay, so I do want to do a few
examples in sort of increasing
00:18:48.230 --> 00:18:50.170
level of difficulty
in how you would
00:18:50.170 --> 00:18:51.420
combine these things together.
00:18:51.420 --> 00:18:55.980
So first of all,
you should remember
00:18:55.980 --> 00:19:00.885
that if you differentiate the
secant function, that's just
00:19:00.885 --> 00:19:03.780
- oh I just realized that
I wanted to say something
00:19:03.780 --> 00:19:06.630
else before - so forget that.
00:19:06.630 --> 00:19:08.060
We'll do that in a second.
00:19:08.060 --> 00:19:10.980
I wanted to make
some general remarks.
00:19:10.980 --> 00:19:17.600
So there's one rule that you
discussed in my absence, which
00:19:17.600 --> 00:19:19.070
is the chain rule.
00:19:19.070 --> 00:19:21.780
And I do want to make
just a couple of remarks
00:19:21.780 --> 00:19:26.160
about the chain rule now to
remind you of what it is,
00:19:26.160 --> 00:19:30.160
and also to present
some consequences.
00:19:30.160 --> 00:19:39.190
So, a little bit of
extra on the chain rule.
00:19:39.190 --> 00:19:43.720
The first thing that I want
say is that we didn't really
00:19:43.720 --> 00:19:46.660
fully explain why it's true.
00:19:46.660 --> 00:19:54.140
And I do want to just
explain it by example, okay?
00:19:54.140 --> 00:19:59.720
So imagine that you have
a function which is, say,
00:19:59.720 --> 00:20:02.000
10x + b.
00:20:02.000 --> 00:20:02.500
All right?
00:20:02.500 --> 00:20:04.980
So y = 10x + b.
00:20:04.980 --> 00:20:09.970
Then obviously, y is changing
10 times as fast as b, right?
00:20:09.970 --> 00:20:18.060
The issue is this number
here, dy/dx, is 10.
00:20:18.060 --> 00:20:20.460
And now if x is a
function of something,
00:20:20.460 --> 00:20:34.290
say t, shifted by some other
constant here, then dx/dt = 5.
00:20:34.290 --> 00:20:38.800
Now all the chain rule is saying
is that if y is going 10 times
00:20:38.800 --> 00:20:44.610
as fast as t, I'm sorry as
x, and x is going 5 times
00:20:44.610 --> 00:20:50.620
as fast as t, then y is
going 50 times as fast as t.
00:20:50.620 --> 00:20:53.530
And algebraically, all
this means is if I plug
00:20:53.530 --> 00:20:57.100
in and substitute, which is
what the composition of the two
00:20:57.100 --> 00:21:04.760
functions amounts to, 10(5t +
a) + b and I multiply it out,
00:21:04.760 --> 00:21:09.200
I get 50t + 10a + b.
00:21:09.200 --> 00:21:11.697
Now these terms don't matter.
00:21:11.697 --> 00:21:13.030
The constant terms don't matter.
00:21:13.030 --> 00:21:14.800
The rate is 50.
00:21:14.800 --> 00:21:17.130
And so the consequence,
if we put them together,
00:21:17.130 --> 00:21:30.170
is that dy/dt =
10*5, which is 50.
00:21:30.170 --> 00:21:31.990
All right, so this
is in a nutshell
00:21:31.990 --> 00:21:33.630
why the chain rule works.
00:21:33.630 --> 00:21:39.450
And why these rates multiply.
00:21:39.450 --> 00:21:42.460
The second thing that I wanted
to say about the chain rule
00:21:42.460 --> 00:21:45.050
is that it has a few
consequences that
00:21:45.050 --> 00:21:47.380
make some of the other
rules a little easier
00:21:47.380 --> 00:21:50.220
to remember or
possibly to avoid.
00:21:50.220 --> 00:21:54.510
The messiest rule
in my humble opinion
00:21:54.510 --> 00:21:59.430
is the quotient rule, which is
kind of a nuisance to remember.
00:21:59.430 --> 00:22:01.150
So let me just
remind you, if you
00:22:01.150 --> 00:22:03.840
take just the reciprocal
of a function,
00:22:03.840 --> 00:22:05.990
and you differentiate
it, there's
00:22:05.990 --> 00:22:08.170
another way of looking at this.
00:22:08.170 --> 00:22:09.720
And it's actually
the way that I use,
00:22:09.720 --> 00:22:12.800
so I want to encourage you to
think about it this way too.
00:22:12.800 --> 00:22:15.670
This is the same as (v^(-1))'.
00:22:15.670 --> 00:22:16.700
.
00:22:16.700 --> 00:22:18.840
And now instead of using
the quotient rule, which
00:22:18.840 --> 00:22:23.840
we could've used, we can
use the chain rule here
00:22:23.840 --> 00:22:29.720
with the power -1, which
works by the power law.
00:22:29.720 --> 00:22:30.960
So what is this equal to?
00:22:30.960 --> 00:22:33.640
This is equal to -v^(-2) v'.
00:22:38.930 --> 00:22:42.730
So here, I've applied the
chain rule rather than
00:22:42.730 --> 00:22:47.370
the quotient rule.
00:22:47.370 --> 00:22:54.060
And similarly, suppose I wanted
to derive the full quotient
00:22:54.060 --> 00:22:54.560
rule.
00:22:54.560 --> 00:22:57.240
Well, now this may
or may not be easier.
00:22:57.240 --> 00:22:59.770
But this is one way of
remembering what's going on.
00:22:59.770 --> 00:23:05.300
If you convert it to uv^(-1)
and you differentiate that,
00:23:05.300 --> 00:23:09.180
now I can use the
product rule on this.
00:23:09.180 --> 00:23:11.970
Of course, I have to use
the chain rule and this rule
00:23:11.970 --> 00:23:13.030
as well.
00:23:13.030 --> 00:23:15.620
So what do I get?
00:23:15.620 --> 00:23:21.519
I get u', the inverse,
plus u, and then I have
00:23:21.519 --> 00:23:22.810
to differentiate the v inverse.
00:23:22.810 --> 00:23:24.490
That's the formula
right up here.
00:23:24.490 --> 00:23:25.290
That's -v^(-2) v'.
00:23:30.300 --> 00:23:33.230
So that's one way of doing it.
00:23:33.230 --> 00:23:35.760
This actually explains
the funny minus sign
00:23:35.760 --> 00:23:38.560
when you differentiate
v in the formula.
00:23:38.560 --> 00:23:41.280
The other formula, the
other way that we did it,
00:23:41.280 --> 00:23:44.370
was by putting this over
a common denominator.
00:23:44.370 --> 00:23:49.330
The common denominator was v^2.
00:23:49.330 --> 00:23:51.580
This comes from this v v^(-2).
00:23:51.580 --> 00:23:54.730
And then the second
term is -uv'.
00:23:57.250 --> 00:24:00.020
And the first term, we have
to multiply by an extra factor
00:24:00.020 --> 00:24:02.190
of v, because we have a
v^2 in the denominator.
00:24:02.190 --> 00:24:07.720
So it's u'v. All right, so
this is the quotient rule as we
00:24:07.720 --> 00:24:11.247
wrote it down in lecture,
and this is just another way
00:24:11.247 --> 00:24:13.580
of remembering it or deriving
it without remembering it,
00:24:13.580 --> 00:24:16.700
if you just remember the chain
rule and the product rule.
00:24:16.700 --> 00:24:19.710
Okay, so you'll find
that in many contexts,
00:24:19.710 --> 00:24:25.910
it's easier to do
one or the other.
00:24:25.910 --> 00:24:29.210
Okay, so now I'm ready to
differentiate the secant
00:24:29.210 --> 00:24:30.990
and a few such functions.
00:24:30.990 --> 00:24:36.200
So we'll do some
examples here here.
00:24:36.200 --> 00:24:39.030
So here's the secant
function, and I
00:24:39.030 --> 00:24:44.820
want to use that formula up
there for the reciprocal.
00:24:44.820 --> 00:24:48.090
This is the way I think of it.
00:24:48.090 --> 00:24:53.150
This is the cosine
function to the power -1.
00:24:53.150 --> 00:24:58.750
And so, the formula
here is just what?
00:24:58.750 --> 00:25:04.030
It's just -(cos
x)^(-2) times -sin x.
00:25:20.280 --> 00:25:22.550
So now this is usually written
in a different fashion,
00:25:22.550 --> 00:25:25.170
so that's why I'm doing
this for a reason actually.
00:25:25.170 --> 00:25:27.810
Which is although there are
several formulas for things,
00:25:27.810 --> 00:25:29.810
with trig functions,
there are usually
00:25:29.810 --> 00:25:31.854
five ways of writing something.
00:25:31.854 --> 00:25:33.520
So I'm writing this
one down so that you
00:25:33.520 --> 00:25:36.780
know what the standard
way of presenting it is.
00:25:36.780 --> 00:25:39.760
So what happens here is
that we have two minus signs
00:25:39.760 --> 00:25:40.300
cancelling.
00:25:40.300 --> 00:25:44.360
And we get sin x / cos^2 x.
00:25:44.360 --> 00:25:46.430
That's a perfectly
acceptable answer,
00:25:46.430 --> 00:25:49.470
but there's a customary
way in which is written.
00:25:49.470 --> 00:25:55.890
It's written (1 / cos
x) (sin x / cos x).
00:25:55.890 --> 00:25:57.530
And then we get rid
of the denominators
00:25:57.530 --> 00:26:00.710
by rewriting it in terms
of secant and tangent,
00:26:00.710 --> 00:26:04.100
so sec x tan x.
00:26:04.100 --> 00:26:07.680
So this is the form
that's generally
00:26:07.680 --> 00:26:11.790
used when you see these
formulas written in textbooks.
00:26:11.790 --> 00:26:14.440
And so you know, you
need to watch out,
00:26:14.440 --> 00:26:16.810
because if you ever want to
use this kind of calculus,
00:26:16.810 --> 00:26:22.840
you'll have not be put off by
all the secants and tangents.
00:26:22.840 --> 00:26:26.830
All right, so getting
slightly more complicated,
00:26:26.830 --> 00:26:28.750
how about if we
differentiate ln(sec x)?
00:26:37.400 --> 00:26:39.560
If you differentiate
the natural log,
00:26:39.560 --> 00:26:49.450
that's just going to
be (sec x)' / sec x.
00:26:49.450 --> 00:26:51.250
And plugging in
the formula that we
00:26:51.250 --> 00:27:00.330
had before, that's sec x tan
x / sec x, which is tan x.
00:27:00.330 --> 00:27:03.850
So this one also has
a very nice form.
00:27:03.850 --> 00:27:07.940
And you might say that this
is kind of an ugly function,
00:27:07.940 --> 00:27:14.120
but the strange thing is that
the natural log was invented
00:27:14.120 --> 00:27:19.030
before the exponential by
a guy named Napier, exactly
00:27:19.030 --> 00:27:21.720
in order to evaluate
functions like this.
00:27:21.720 --> 00:27:25.930
These are the functions that
people cared about a lot,
00:27:25.930 --> 00:27:28.890
because they were
used in navigation.
00:27:28.890 --> 00:27:32.640
You wanted to multiply
sines and cosines together
00:27:32.640 --> 00:27:34.030
to do navigation.
00:27:34.030 --> 00:27:38.794
And the multiplication he
encoded using a logarithm.
00:27:38.794 --> 00:27:40.710
So these were invented
long before people even
00:27:40.710 --> 00:27:42.794
knew about exponents.
00:27:42.794 --> 00:27:44.460
And it was a surprise,
actually, that it
00:27:44.460 --> 00:27:46.100
was connected to exponents.
00:27:46.100 --> 00:27:48.650
So the natural log was
invented before the log base 10
00:27:48.650 --> 00:27:52.650
and everything else, exactly
for this kind of purpose.
00:27:52.650 --> 00:27:54.550
Anyway, so this is
a nice function,
00:27:54.550 --> 00:27:58.010
which was very important,
so that your ships wouldn't
00:27:58.010 --> 00:28:03.770
crash into the reef.
00:28:03.770 --> 00:28:05.570
Okay, let's continue here.
00:28:05.570 --> 00:28:08.870
So there's another
kind of function
00:28:08.870 --> 00:28:10.490
that I want to discuss with you.
00:28:10.490 --> 00:28:12.460
And these are the
kinds in which there's
00:28:12.460 --> 00:28:19.380
a choice as to which of
these rules to apply.
00:28:19.380 --> 00:28:25.130
And I'll just give a
couple of examples of that.
00:28:25.130 --> 00:28:27.620
There usually is a
better and a worse way,
00:28:27.620 --> 00:28:38.430
so let me illustrate that.
00:28:38.430 --> 00:28:41.120
Okay, yet another example.
00:28:41.120 --> 00:28:43.830
I hope you've seen
some of these before.
00:28:43.830 --> 00:28:46.660
Say (x^10 + 8x)^6.
00:28:51.010 --> 00:28:52.990
So it's a little bit more
complicated than what
00:28:52.990 --> 00:29:00.330
we had before, because there
were several more symbols here.
00:29:00.330 --> 00:29:03.210
So what should we
do at this point?
00:29:03.210 --> 00:29:06.210
There's one choice which
I claim is a bad idea,
00:29:06.210 --> 00:29:10.810
and that is to expand
this out to the 6th power.
00:29:10.810 --> 00:29:13.530
That's a bad idea,
because it's very long.
00:29:13.530 --> 00:29:15.990
And then your answer
will also be very long.
00:29:15.990 --> 00:29:19.521
It will fill the entire
exam paper, for instance.
00:29:19.521 --> 00:29:20.020
Yeah?
00:29:20.020 --> 00:29:21.380
STUDENT: Can you
use the chain rule?
00:29:21.380 --> 00:29:21.970
PROFESSOR: Chain rule.
00:29:21.970 --> 00:29:22.580
That's it.
00:29:22.580 --> 00:29:23.500
We use the chain rule.
00:29:23.500 --> 00:29:26.620
So fortunately, this
is relatively easy
00:29:26.620 --> 00:29:27.620
using the chain rule.
00:29:27.620 --> 00:29:30.790
We just think of this box
as being the function.
00:29:30.790 --> 00:29:34.560
And we take 6 times
this guy to the 5th,
00:29:34.560 --> 00:29:37.570
times the derivative of this
guy, which is 10x^9 + 8.
00:29:41.500 --> 00:29:43.910
And this is, filling
this in, it's x^10 + 8x.
00:29:43.910 --> 00:29:46.140
And that's it.
00:29:46.140 --> 00:29:50.270
That's all you need to do
differentiate things like this.
00:29:50.270 --> 00:29:55.140
The chain rule is
very effective.
00:29:55.140 --> 00:29:59.864
STUDENT: [INAUDIBLE]
00:29:59.864 --> 00:30:01.280
PROFESSOR: That's
a good question.
00:30:01.280 --> 00:30:04.330
So I'm not really willing
to answer too many questions
00:30:04.330 --> 00:30:07.500
about what's going
to be on the exam.
00:30:07.500 --> 00:30:09.070
But the question
that was just asked
00:30:09.070 --> 00:30:13.200
is exactly the kind of question
I'm very happy to answer.
00:30:13.200 --> 00:30:18.350
Okay, the question was,
in what form is-- what
00:30:18.350 --> 00:30:20.090
form is an acceptable answer?
00:30:20.090 --> 00:30:23.560
Now in real life, that is
a really serious question.
00:30:23.560 --> 00:30:25.200
When you ask a
computer a question
00:30:25.200 --> 00:30:28.930
and it gives you 500
million sheets of printout,
00:30:28.930 --> 00:30:31.380
it's useless.
00:30:31.380 --> 00:30:33.960
And you really care what
form answers are in,
00:30:33.960 --> 00:30:35.730
and indeed, somebody
might really
00:30:35.730 --> 00:30:39.000
care what this thing
to the 6th power is,
00:30:39.000 --> 00:30:42.090
and then you would be forced
to discuss things in terms
00:30:42.090 --> 00:30:46.110
of that other functional form.
00:30:46.110 --> 00:30:50.410
For the purposes of this
exam, this is okay form.
00:30:50.410 --> 00:30:54.490
And, in fact, any correct
form is an okay form.
00:30:54.490 --> 00:30:57.770
I recommend strongly that you
not try to simplify things
00:30:57.770 --> 00:30:59.700
unless we tell you to.
00:30:59.700 --> 00:31:04.860
Sometimes it will be to your
advantage to simplify things.
00:31:04.860 --> 00:31:08.010
Sometimes we'll say simplify.
00:31:08.010 --> 00:31:10.390
It takes a good
deal of experience
00:31:10.390 --> 00:31:13.121
to know when it's really worth
it to simplify expressions.
00:31:13.121 --> 00:31:13.620
Yes?
00:31:13.620 --> 00:31:19.530
STUDENT: [INAUDIBLE]
00:31:19.530 --> 00:31:23.590
PROFESSOR: Right, so
turning to this example.
00:31:23.590 --> 00:31:25.520
The question is what
is this derivative?
00:31:25.520 --> 00:31:27.240
And here's an answer.
00:31:27.240 --> 00:31:29.500
That's the end of the problem.
00:31:29.500 --> 00:31:31.810
This is a more customary form.
00:31:31.810 --> 00:31:37.160
But this is answer is okay.
00:31:37.160 --> 00:31:38.610
Same issue.
00:31:38.610 --> 00:31:40.970
That's exactly the point.
00:31:40.970 --> 00:31:41.660
Yes?
00:31:41.660 --> 00:31:51.460
STUDENT: [INAUDIBLE]
00:31:51.460 --> 00:31:59.032
PROFESSOR: The question is,
do you have to show the work?
00:31:59.032 --> 00:32:00.240
Do you have to show the work?
00:32:00.240 --> 00:32:04.870
Well if I ask you
what is d/dx of sec x,
00:32:04.870 --> 00:32:06.650
then if you wrote
down this answer
00:32:06.650 --> 00:32:09.510
or you wrote down this
answer showing no work,
00:32:09.510 --> 00:32:11.200
that would be acceptable.
00:32:11.200 --> 00:32:15.950
If the question was derive
the formula for this
00:32:15.950 --> 00:32:18.650
from the formula for the
derivative of the cosine
00:32:18.650 --> 00:32:21.160
or something like that, then
it would not be acceptable.
00:32:21.160 --> 00:32:24.340
You'd have to carry
out this arithmetic.
00:32:24.340 --> 00:32:28.470
So, in other words,
typically this
00:32:28.470 --> 00:32:32.290
will come up, for instance,
in various contexts.
00:32:32.290 --> 00:32:34.830
You just basically have
to follow directions.
00:32:34.830 --> 00:32:35.330
Yes?
00:32:35.330 --> 00:32:41.424
STUDENT: [INAUDIBLE]
00:32:41.424 --> 00:32:43.090
PROFESSOR: The next
question is, are you
00:32:43.090 --> 00:32:44.465
expected to be
able to prove what
00:32:44.465 --> 00:32:46.180
the derivative of
the sine function is?
00:32:46.180 --> 00:32:49.580
The short answer to that is yes.
00:32:49.580 --> 00:32:51.630
But I will be getting
to that when I discuss
00:32:51.630 --> 00:32:54.240
the rest of the material here.
00:32:54.240 --> 00:32:58.430
We're almost there.
00:32:58.430 --> 00:33:02.640
Okay, so let me just
finish these examples
00:33:02.640 --> 00:33:04.880
with one last one.
00:33:04.880 --> 00:33:06.880
And then we'll talk
about this question
00:33:06.880 --> 00:33:10.630
of things like the derivative
of the sine function,
00:33:10.630 --> 00:33:12.060
and deriving it.
00:33:12.060 --> 00:33:15.620
So the last example that I'd
like to write down is the one
00:33:15.620 --> 00:33:18.940
that I promised you
in the first lecture,
00:33:18.940 --> 00:33:26.172
namely to differentiate
e^(x tan^(-1) x).
00:33:26.172 --> 00:33:28.380
Basically you're supposed
to be able to differentiate
00:33:28.380 --> 00:33:29.350
any function.
00:33:29.350 --> 00:33:32.390
So this is the one that we
mentioned at the beginning.
00:33:32.390 --> 00:33:34.130
So here it is.
00:33:34.130 --> 00:33:37.280
Let's do it.
00:33:37.280 --> 00:33:38.170
So what is it?
00:33:38.170 --> 00:33:45.742
Well, it's just equal to
- I have to differentiate.
00:33:45.742 --> 00:33:47.200
I have to use the
chain rule - it's
00:33:47.200 --> 00:33:52.930
equal to the exponential
times the derivative
00:33:52.930 --> 00:33:58.200
of this expression here.
00:33:58.200 --> 00:33:59.260
That's the chain rule.
00:33:59.260 --> 00:34:01.700
That's the first step.
00:34:01.700 --> 00:34:06.440
And now I have to apply
the product rule here.
00:34:06.440 --> 00:34:10.820
So I have e^(x tan^(-1) x).
00:34:10.820 --> 00:34:15.809
And I differentiate the first
factor, so I get tan^(-1) x.
00:34:15.809 --> 00:34:17.600
Add to it what happens
when I differentiate
00:34:17.600 --> 00:34:19.670
the second factor,
leaving alone the x.
00:34:19.670 --> 00:34:21.450
So that's x / (1+x^2).
00:34:24.310 --> 00:34:26.300
And that's it.
00:34:26.300 --> 00:34:28.780
That's the end of the problem.
00:34:28.780 --> 00:34:30.590
It wasn't that hard.
00:34:30.590 --> 00:34:35.330
Of course, it requires you
to remember all of the rules,
00:34:35.330 --> 00:34:37.300
and a lot of formulas
underlying them.
00:34:37.300 --> 00:34:39.560
So that's consistent with
what I just told you.
00:34:39.560 --> 00:34:42.060
I told you that you
wanted to know this.
00:34:42.060 --> 00:34:44.740
I told you that you needed
to know this product rule,
00:34:44.740 --> 00:34:50.419
and that you needed to
know the chain rule.
00:34:50.419 --> 00:34:51.960
And I guess there
was one more thing,
00:34:51.960 --> 00:34:55.260
the derivative of e^x
came into play there.
00:34:55.260 --> 00:34:59.040
So of these formulas,
we used four of them
00:34:59.040 --> 00:35:03.810
in this one calculation.
00:35:03.810 --> 00:35:15.880
Okay, so now what other things
did we talk about in Unit One?
00:35:15.880 --> 00:35:23.590
So the main other thing
that we talked about
00:35:23.590 --> 00:35:33.120
was the definition
of a derivative.
00:35:33.120 --> 00:35:40.020
And also there
was sort of a goal
00:35:40.020 --> 00:35:51.050
which was to get to the
meaning of the derivative.
00:35:51.050 --> 00:35:56.520
So these are things - so we
had a couple of ways of looking
00:35:56.520 --> 00:35:59.170
at it, or at least
a couple that I'm
00:35:59.170 --> 00:36:01.780
going to emphasize right now.
00:36:01.780 --> 00:36:06.270
But first, let me remind
you what the formula is.
00:36:06.270 --> 00:36:13.900
The derivative is the limit
as delta x goes to 0 of (f(x +
00:36:13.900 --> 00:36:19.040
delta x) - f(x)) / delta x.
00:36:19.040 --> 00:36:22.430
So that's it, and
this is certainly
00:36:22.430 --> 00:36:25.640
going to be a
central focus here.
00:36:25.640 --> 00:36:29.600
And you want to be able
to recognize this formula
00:36:29.600 --> 00:36:42.760
in a number of ways.
00:36:42.760 --> 00:36:44.520
So, how do we use this?
00:36:44.520 --> 00:36:48.950
Well one thing we did
was we calculated a bunch
00:36:48.950 --> 00:36:51.450
of these rates of change.
00:36:51.450 --> 00:36:53.580
In fact, more or less,
they're the ones which
00:36:53.580 --> 00:36:55.760
are written right over here.
00:36:55.760 --> 00:36:57.210
This list of functions here.
00:36:57.210 --> 00:37:01.470
Now, which ones did we
start out with just straight
00:37:01.470 --> 00:37:03.800
from the definition here?
00:37:03.800 --> 00:37:04.840
Which of these things?
00:37:04.840 --> 00:37:06.215
There were a whole
bunch of them.
00:37:06.215 --> 00:37:09.180
So we started out
with a function 1/x.
00:37:09.180 --> 00:37:11.530
We did x^n.
00:37:11.530 --> 00:37:14.530
We did sine x.
00:37:14.530 --> 00:37:16.880
We did cosine x.
00:37:16.880 --> 00:37:19.205
Now there was a little
bit of subtlety with sine
00:37:19.205 --> 00:37:21.110
x and cosine x.
00:37:21.110 --> 00:37:25.210
We got them using
something else.
00:37:25.210 --> 00:37:26.880
We didn't quite get
them all the way.
00:37:26.880 --> 00:37:31.790
We got them using
the case x = 0.
00:37:31.790 --> 00:37:34.530
We got them from the
derivative at x = 0,
00:37:34.530 --> 00:37:37.680
we got the formulas for the
derivatives of sine and cosine.
00:37:37.680 --> 00:37:40.890
But that was an argument
which involved plugging in sin
00:37:40.890 --> 00:37:44.460
(x + delta x), and
running through.
00:37:44.460 --> 00:37:45.840
So that's one example.
00:37:45.840 --> 00:37:50.630
We also did a^x.
00:37:50.630 --> 00:37:53.270
And that may be it.
00:37:53.270 --> 00:37:58.350
Oh yeah, I think
that's about it.
00:37:58.350 --> 00:38:00.450
That may be about it.
00:38:00.450 --> 00:38:00.950
No.
00:38:00.950 --> 00:38:01.620
It isn't.
00:38:01.620 --> 00:38:03.910
Okay, so let me make a
connection here which you
00:38:03.910 --> 00:38:07.770
probably haven't yet made, which
is that we did it for (u v)'.
00:38:10.520 --> 00:38:15.690
And we also did it for (u / v)'.
00:38:15.690 --> 00:38:17.460
So sorry, I shouldn't
write primes,
00:38:17.460 --> 00:38:20.500
because that's not consistent
with the claim there.
00:38:20.500 --> 00:38:24.900
I differentiated the product;
I differentiated the quotient
00:38:24.900 --> 00:38:28.110
using the same delta x notation.
00:38:28.110 --> 00:38:32.760
I guess I forgot that because I
wasn't there when it happened.
00:38:32.760 --> 00:38:36.550
So look, these are the
ones that you do by this.
00:38:36.550 --> 00:38:39.310
And, of course, you might have
to reduce them to other things.
00:38:39.310 --> 00:38:42.190
These involve using
something else.
00:38:42.190 --> 00:38:46.610
This one involves using the
slope of this function at 0,
00:38:46.610 --> 00:38:48.600
just the way the sine
and the cosine did.
00:38:48.600 --> 00:38:52.410
This one involves the slopes
of the individual functions, u
00:38:52.410 --> 00:38:54.827
and v. And this one also
involves the individual--
00:38:54.827 --> 00:38:56.410
So, in other words,
it doesn't get you
00:38:56.410 --> 00:38:58.390
all the way through
to the end, but it's
00:38:58.390 --> 00:39:03.010
expressed in terms of something
simpler in each of these cases.
00:39:03.010 --> 00:39:05.840
And I could go on.
00:39:05.840 --> 00:39:09.280
We didn't do these in class,
but you're certainly--
00:39:09.280 --> 00:39:12.170
e^x is a perfectly okay
one on one of the exams.
00:39:12.170 --> 00:39:14.561
We ask you for 1/x^2.
00:39:14.561 --> 00:39:16.310
In other words, I'm
not claiming that it's
00:39:16.310 --> 00:39:18.380
going to be one on this
list, but it certainly
00:39:18.380 --> 00:39:19.671
can be any one of these.
00:39:19.671 --> 00:39:21.170
But we're not going
to ask you to go
00:39:21.170 --> 00:39:26.380
all the way through to the
beginning in these formulas.
00:39:26.380 --> 00:39:28.940
There are also some fundamental
limits that I certainly
00:39:28.940 --> 00:39:31.210
want you to know about.
00:39:31.210 --> 00:39:34.680
And these you can
derive in reverse.
00:39:34.680 --> 00:39:58.880
So I will describe that now.
00:39:58.880 --> 00:40:06.800
So let me also emphasize
the following thing: I want
00:40:06.800 --> 00:40:18.590
to read this backwards now.
00:40:18.590 --> 00:40:21.370
This is the theme from the
very beginning of this lecture.
00:40:21.370 --> 00:40:25.210
Namely, if you're
given the function f,
00:40:25.210 --> 00:40:27.811
you can figure out its
derivative by this formula
00:40:27.811 --> 00:40:28.310
here.
00:40:28.310 --> 00:40:29.860
That is the formula for
this in terms of what's
00:40:29.860 --> 00:40:30.920
on the right hand side.
00:40:30.920 --> 00:40:34.400
On the other hand,
you can also use
00:40:34.400 --> 00:40:45.280
the formula in that
direction, and if you
00:40:45.280 --> 00:40:48.310
know the slope of something,
you can figure out
00:40:48.310 --> 00:40:49.170
what the limit is.
00:40:49.170 --> 00:40:54.570
For example, I'll use
the letter x here,
00:40:54.570 --> 00:40:56.040
even though it's cheating.
00:40:56.040 --> 00:40:59.530
Maybe I'll call it delta
x so it's clearer to you.
00:40:59.530 --> 00:41:06.900
Maybe I'll call it u.
00:41:06.900 --> 00:41:10.370
Suppose you look
at this limit here.
00:41:10.370 --> 00:41:14.800
Well, I claim that you
should recognize that is
00:41:14.800 --> 00:41:19.870
the derivative with respect to
u of the function e^u at u = 0,
00:41:19.870 --> 00:41:22.660
which of course we know to be 1.
00:41:22.660 --> 00:41:25.420
So this is reading this
formula in reverse.
00:41:25.420 --> 00:41:27.940
It's recognizing that
one of these limits -
00:41:27.940 --> 00:41:35.160
let me rewrite this again
here - one of these so-called
00:41:35.160 --> 00:41:39.390
difference quotient
limits is a derivative.
00:41:39.390 --> 00:41:42.190
And since we know a formula
for that derivative,
00:41:42.190 --> 00:41:49.940
we can evaluate it.
00:41:49.940 --> 00:41:54.150
And lastly, there's
one other type of thing
00:41:54.150 --> 00:41:57.550
which I think you should know.
00:41:57.550 --> 00:41:59.767
These are the ones you do
with difference quotients.
00:41:59.767 --> 00:42:01.350
There are also other
formulas that you
00:42:01.350 --> 00:42:03.000
want to be able to derive.
00:42:03.000 --> 00:42:19.740
You want to be able
to derive formulas
00:42:19.740 --> 00:42:27.670
by implicit differentiation.
00:42:27.670 --> 00:42:30.220
In other words,
the basic idea is
00:42:30.220 --> 00:42:32.150
to take whatever
equation you've got
00:42:32.150 --> 00:42:36.560
and simplify it as
much as possible,
00:42:36.560 --> 00:42:41.260
without insisting
that you solve for y.
00:42:41.260 --> 00:42:44.080
That's not necessarily
the most appropriate way
00:42:44.080 --> 00:42:45.630
to get the rate of change.
00:42:45.630 --> 00:42:51.910
The much simpler
formula is sin y = x.
00:42:51.910 --> 00:42:59.780
And that one is easier to
differentiate implicitly.
00:42:59.780 --> 00:43:02.900
So I should say, do
this kind of thing.
00:43:02.900 --> 00:43:05.550
So that's, if you like,
a typical derivation
00:43:05.550 --> 00:43:08.390
that you might see.
00:43:08.390 --> 00:43:13.070
And then there's one last type
of problem that you'll face,
00:43:13.070 --> 00:43:21.590
and it's the other thing
that I claim we discussed.
00:43:21.590 --> 00:43:26.580
And it goes all the way
back to the first lecture.
00:43:26.580 --> 00:43:33.700
So the last thing that we'll be
talking about is tangent lines.
00:43:33.700 --> 00:43:34.200
All right?
00:43:34.200 --> 00:43:38.760
The geometric point of
view of a derivative.
00:43:38.760 --> 00:43:41.900
And we'll be doing more
of this in next the unit.
00:43:41.900 --> 00:43:44.850
So first of all,
you'll be expected
00:43:44.850 --> 00:43:52.380
to be able to compute
the tangent line.
00:43:52.380 --> 00:43:56.400
That's often fairly
straightforward.
00:43:56.400 --> 00:44:03.100
And the second thing
is to graph y' ,
00:44:03.100 --> 00:44:07.370
the derivative of a function.
00:44:07.370 --> 00:44:09.350
And the third thing,
which I'm going
00:44:09.350 --> 00:44:11.550
to throw in here,
because I regard it
00:44:11.550 --> 00:44:14.900
in a sort of geometric
vein, although it's got
00:44:14.900 --> 00:44:16.690
an analytical aspect to it.
00:44:16.690 --> 00:44:18.870
So this is a picture.
00:44:18.870 --> 00:44:20.710
This is a computation.
00:44:20.710 --> 00:44:23.080
And if you combine
the two together,
00:44:23.080 --> 00:44:24.270
you get something else.
00:44:24.270 --> 00:44:37.870
And so this is to recognize
differentiable functions.
00:44:37.870 --> 00:44:40.190
Alright, so how do you do this?
00:44:40.190 --> 00:44:43.600
Well, we really only
have one way of doing it.
00:44:43.600 --> 00:44:54.580
We're going to check the
left and right tangents.
00:44:54.580 --> 00:44:59.450
They must be equal.
00:44:59.450 --> 00:45:04.280
So again, this is
a property that you
00:45:04.280 --> 00:45:06.830
should be familiar with
from some of your exercises.
00:45:06.830 --> 00:45:09.660
And the idea is simply, that
if the tangent line exists,
00:45:09.660 --> 00:45:14.770
it's the same from the
right and from the left.
00:45:14.770 --> 00:45:20.200
Okay, now I'm going to
just do one example here
00:45:20.200 --> 00:45:25.450
from this sort of
qualitative sketching skill
00:45:25.450 --> 00:45:27.289
to give you an example here.
00:45:27.289 --> 00:45:28.830
And what I'm going
to do is I'm going
00:45:28.830 --> 00:45:34.750
to draw a graph of a
function like this.
00:45:34.750 --> 00:45:38.490
And what I want to
do underneath is draw
00:45:38.490 --> 00:45:41.600
the graph of the derivative.
00:45:41.600 --> 00:45:45.900
So this is the
function y = f(x),
00:45:45.900 --> 00:45:48.350
and here I'm going to draw
the graph of the function y =
00:45:48.350 --> 00:45:56.490
f'(x) right underneath it.
00:45:56.490 --> 00:46:00.330
So now, let's think about what
it's supposed to look like.
00:46:00.330 --> 00:46:05.840
And the one step that you need
to make in order to do this,
00:46:05.840 --> 00:46:08.660
is to draw a few tangent lines.
00:46:08.660 --> 00:46:13.210
I'm just going to
draw one down here.
00:46:13.210 --> 00:46:18.730
And I'm going to
draw one up here.
00:46:18.730 --> 00:46:22.740
Now, the tangent
lines here - notice
00:46:22.740 --> 00:46:27.266
that the slope of these
tangent lines are all positive.
00:46:27.266 --> 00:46:28.640
So everything I
draw down here is
00:46:28.640 --> 00:46:33.880
going to be above the x-axis.
00:46:33.880 --> 00:46:36.190
Furthermore, as I go
further to the left,
00:46:36.190 --> 00:46:37.830
they get steeper and steeper.
00:46:37.830 --> 00:46:39.500
So they're getting
higher and higher.
00:46:39.500 --> 00:46:44.020
So the function is
coming down like this.
00:46:44.020 --> 00:46:45.350
It starts up there.
00:46:45.350 --> 00:46:50.570
Maybe I'll draw it in green
to illustrate the graph here.
00:46:50.570 --> 00:46:56.910
So that's this function here.
00:46:56.910 --> 00:46:59.750
As we get farther out, it's
getting flatter and flatter.
00:46:59.750 --> 00:47:06.270
So it's leveling off, but
above the axis like that.
00:47:06.270 --> 00:47:08.280
So one of the
things to emphasize
00:47:08.280 --> 00:47:10.830
is, you should not
expect the derivative
00:47:10.830 --> 00:47:12.090
to look like the function.
00:47:12.090 --> 00:47:13.320
It's totally different.
00:47:13.320 --> 00:47:17.280
It's keeping track at each
point of its tangent line.
00:47:17.280 --> 00:47:19.780
On the other hand, you should
get some kind of physical feel
00:47:19.780 --> 00:47:23.619
for it, and we'll be practicing
this more in the next unit.
00:47:23.619 --> 00:47:25.660
So let me give you an
example of a function which
00:47:25.660 --> 00:47:27.800
does exactly this.
00:47:27.800 --> 00:47:33.240
And it's the function y = ln x.
00:47:33.240 --> 00:47:38.560
If you differentiate
it, you get y' = 1/x.
00:47:38.560 --> 00:47:44.630
And this plot above is, roughly
speaking, the logarithm.
00:47:44.630 --> 00:47:50.230
And this plot underneath
is the function 1/x.
00:47:50.230 --> 00:47:53.230
We still have time
for one question.
00:47:53.230 --> 00:47:58.580
And so, fire away.
00:47:58.580 --> 00:48:03.207
Yes?
00:48:03.207 --> 00:48:04.040
STUDENT: [INAUDIBLE]
00:48:04.040 --> 00:48:05.580
PROFESSOR: The
question is, can you
00:48:05.580 --> 00:48:09.770
show how you derive the
inverse tangent of x.
00:48:09.770 --> 00:48:13.350
So that's in a lecture.
00:48:13.350 --> 00:48:17.060
I'm happy to do it right
now, but it's going
00:48:17.060 --> 00:48:20.420
to take me a whole two minutes.
00:48:20.420 --> 00:48:27.560
So, here's how you do
it. y = tan^(-1) x.
00:48:27.560 --> 00:48:30.230
And now this is hopeless
to differentiate,
00:48:30.230 --> 00:48:34.720
so I rewrite it as tan y = x.
00:48:34.720 --> 00:48:38.440
And now I have to
differentiate that.
00:48:38.440 --> 00:48:42.360
So when I
differentiate it, I get
00:48:42.360 --> 00:48:44.260
the derivative of
tan y with respect
00:48:44.260 --> 00:48:46.560
to x-- with respect to y.
00:48:46.560 --> 00:48:48.210
That's 1 / (1 + y^2) times y'.
00:48:51.120 --> 00:48:52.850
So this is a hard step.
00:48:52.850 --> 00:48:53.930
That's the chain rule.
00:48:53.930 --> 00:48:55.860
And on the left side I get 1.
00:48:55.860 --> 00:48:57.650
So I'm doing this
super fast because we
00:48:57.650 --> 00:49:00.720
have thirty seconds left.
00:49:00.720 --> 00:49:02.840
But this is the hard
step right here.
00:49:02.840 --> 00:49:04.900
And it needs for you
to know that d/dy tan
00:49:04.900 --> 00:49:14.432
y is equal to one over-- Oh,
bad bad bad, secant squared.
00:49:14.432 --> 00:49:22.810
I was ahead of myself so fast.
00:49:22.810 --> 00:49:24.920
So here's the identity.
00:49:24.920 --> 00:49:28.500
So you need have
known this in advance.
00:49:28.500 --> 00:49:30.740
And that's the input
into this equation.
00:49:30.740 --> 00:49:44.000
So now, what we have is
that y' = 1 / sec^2 y y,
00:49:44.000 --> 00:49:51.380
which is the same
thing as cos^2 y.
00:49:51.380 --> 00:49:54.170
Now, the last bit
of the problem is
00:49:54.170 --> 00:49:57.930
to rewrite this in terms of x.
00:49:57.930 --> 00:50:02.664
And that you have to do
with a right triangle.
00:50:02.664 --> 00:50:05.618
If this is x and this
is 1, then the angle
00:50:05.618 --> 00:50:09.416
is y, because the
tangent of y is x.
00:50:09.416 --> 00:50:14.902
So this expresses the fact
that the tangent of y is x.
00:50:14.902 --> 00:50:18.700
And then the hypotenuse is
the square root of 1 + x^2.
00:50:21.654 --> 00:50:27.140
And so the cosine is
1 divided by that.
00:50:27.140 --> 00:50:30.938
So this thing is 1 divided by
the square root of 1 + x^2,
00:50:30.938 --> 00:50:36.424
the quantity squared.
00:50:36.424 --> 00:50:40.644
So, and then the last little bit
here, since I'm racing along,
00:50:40.644 --> 00:50:45.286
is that it's 1 / (1 + x^2),
which I incorrectly wrote over
00:50:45.286 --> 00:50:46.130
here.
00:50:46.130 --> 00:50:48.662
Okay, so good luck on the test.
00:50:48.662 --> 00:50:50.756
See you tomorrow.