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PROFESSOR: Right now, we're
finishing up with the first
10
00:00:25,93 --> 00:00:31,89
unit, and I'd like to continue
in this lecture, lecture seven,
11
00:00:31,89 --> 00:00:45,51
with some final remarks
about exponents.
12
00:00:45,51 --> 00:00:49,95
So what I'd like to do is just
review something that I did
13
00:00:49,95 --> 00:00:53,19
quickly last time, and make a
few philosophical
14
00:00:53,19 --> 00:00:53,92
remarks about it.
15
00:00:53,92 --> 00:00:57,88
I think that the steps involved
were maybe a little tricky, and
16
00:00:57,88 --> 00:01:00,68
so I'd like to go through
it one more time.
17
00:01:00,68 --> 00:01:03,02
Remember, that we were
talking about this
18
00:01:03,02 --> 00:01:05,37
number ak, which is (1
19
00:01:05,37 --> 00:01:05,51
1/k)^k.
20
00:01:05,51 --> 00:01:08,27
21
00:01:08,27 --> 00:01:11,82
And what we showed was that
the limit as k goes to
22
00:01:11,82 --> 00:01:16,92
infinity of ak was e.
23
00:01:16,92 --> 00:01:21,29
So the first thing that I'd
like to do is just explain the
24
00:01:21,29 --> 00:01:27,98
proof a little bit more clearly
than I did last time with fewer
25
00:01:27,98 --> 00:01:31,83
symbols, or at least with this
abbreviation of the symbol
26
00:01:31,83 --> 00:01:35,05
here, to show you what
we actually did.
27
00:01:35,05 --> 00:01:43,79
So I'll just remind you of what
we did last time, and the first
28
00:01:43,79 --> 00:01:46,95
observation was to check,
rather than the limit of
29
00:01:46,95 --> 00:01:49,55
this function, but to
take the ln first.
30
00:01:49,55 --> 00:01:52,17
And this is typically what's
done when you have an
31
00:01:52,17 --> 00:01:54,81
exponential, when you
have an exponent.
32
00:01:54,81 --> 00:01:59,31
And what we found was that
the limit here was 1
33
00:01:59,31 --> 00:02:03,5
as k goes to infinity.
34
00:02:03,5 --> 00:02:05,16
So last time, this
is what we did.
35
00:02:05,16 --> 00:02:08,49
And I just wanted to be
careful and show you exactly
36
00:02:08,49 --> 00:02:09,83
what the next step is.
37
00:02:09,83 --> 00:02:14,36
If you exponentiate this fact;
you take e to this power,
38
00:02:14,36 --> 00:02:21,23
that's going to tend to
e ^ 1, which is just e.
39
00:02:21,23 --> 00:02:26,57
And then, we just observe
that this is the same as ak.
40
00:02:26,57 --> 00:02:32,18
So the basic ingredient
here is that e ^ ln a = a.
41
00:02:32,18 --> 00:02:36,99
That's because the ln function
is the inverse of the
42
00:02:36,99 --> 00:02:38,03
exponential function.
43
00:02:38,03 --> 00:02:38,86
Yes, question?
44
00:02:38,86 --> 00:02:54,19
STUDENT: [INAUDIBLE]
45
00:02:54,19 --> 00:02:59,42
PROFESSOR: So the question was,
wouldn't the log of this be 0
46
00:02:59,42 --> 00:03:01,53
because ak is tending to 1.
47
00:03:01,53 --> 00:03:03,83
But ak isn't tending to 1.
48
00:03:03,83 --> 00:03:06,64
Who said it was?
49
00:03:06,64 --> 00:03:08,89
If you take the logarithm,
which is what we did last
50
00:03:08,89 --> 00:03:12,88
time, logarithm of ak is
indeed k(ln(1 + 1/k)).
51
00:03:12,88 --> 00:03:17,03
52
00:03:17,03 --> 00:03:18,52
That does not tend to 0.
53
00:03:18,52 --> 00:03:22,61
This part of it tends to 0, and
this part tends to infinity.
54
00:03:22,61 --> 00:03:26,46
And they balance each
other, 0 times infinity.
55
00:03:26,46 --> 00:03:29,18
We don't really know yet from
this expression, in fact we did
56
00:03:29,18 --> 00:03:32,63
some cleverness with limits and
derivatives, to figure
57
00:03:32,63 --> 00:03:33,23
out this limit.
58
00:03:33,23 --> 00:03:34,28
It was a very subtle thing.
59
00:03:34,28 --> 00:03:37,56
It turned out to be 1.
60
00:03:37,56 --> 00:03:38,85
All right?
61
00:03:38,85 --> 00:03:41,74
Now, the thing that I'd like to
say - I'm sorry I'm going to
62
00:03:41,74 --> 00:03:46,1
erase this aside here - but you
need to go back to your notes
63
00:03:46,1 --> 00:03:49,12
and remember that this is
what we did last time.
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00:03:49,12 --> 00:03:51,97
Because I want to have room for
the next comment that I want to
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00:03:51,97 --> 00:03:54,42
make on this little
blackboard here.
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00:03:54,42 --> 00:03:59,66
What we just derived was this
property here, but I made a
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00:03:59,66 --> 00:04:03
claim yesterday, and I just
want to emphasize it again so
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00:04:03 --> 00:04:07,83
that we realized what it
is that we're doing.
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00:04:07,83 --> 00:04:08,91
I looked at this backwards.
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00:04:08,91 --> 00:04:11,24
One way you can think of this
is we're evaluating this
71
00:04:11,24 --> 00:04:13
limit and getting an answer.
72
00:04:13 --> 00:04:16,74
But all equalities can be
read both directions.
73
00:04:16,74 --> 00:04:21,3
And we can write it the other
way: e = the limit, as k goes
74
00:04:21,3 --> 00:04:26,38
to infinity, of this
expression here.
75
00:04:26,38 --> 00:04:28,64
So that's just the same thing.
76
00:04:28,64 --> 00:04:31,32
And if we read it backwards,
what we're saying is that this
77
00:04:31,32 --> 00:04:35,7
limit is a formula for e.
78
00:04:35,7 --> 00:04:38,44
So this is very typical
of mathematics.
79
00:04:38,44 --> 00:04:40,7
You want to always reverse your
perspective all the time.
80
00:04:40,7 --> 00:04:45,17
Equations work both ways, and
in this case, we have two
81
00:04:45,17 --> 00:04:46,3
different things here.
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00:04:46,3 --> 00:04:50,55
This e was what we defined as
the base, which when you graph
83
00:04:50,55 --> 00:04:54,51
e ^ x, you get slope 1 at 0.
84
00:04:54,51 --> 00:04:56,85
And then it turns out to be
equal to this limit, which we
85
00:04:56,85 --> 00:04:59,06
can calculate numerically.
86
00:04:59,06 --> 00:05:02,1
If you do this on your
calculators, you, of course,
87
00:05:02,1 --> 00:05:05,53
will have a way of programming
in this number and
88
00:05:05,53 --> 00:05:07,29
evaluating it for each k.
89
00:05:07,29 --> 00:05:10,88
And you'll have another button
available to evaluate this one.
90
00:05:10,88 --> 00:05:13,39
So another way of saying it is
it that there's a relationship
91
00:05:13,39 --> 00:05:14,64
between these two things.
92
00:05:14,64 --> 00:05:19,21
And all of Calculus is a matter
of getting these relationships.
93
00:05:19,21 --> 00:05:21,86
So we can look at these things
in several different ways.
94
00:05:21,86 --> 00:05:25,17
And indeed, that's what we're
going to be doing at least at
95
00:05:25,17 --> 00:05:27,04
the end of today in talking
about derivatives.
96
00:05:27,04 --> 00:05:30,23
A lot of times when we talk
about derivative, we're trying
97
00:05:30,23 --> 00:05:34,78
to look at them from several
perspectives at once.
98
00:05:34,78 --> 00:05:37,58
Okay, so I have to keep on
going with exponents, because
99
00:05:37,58 --> 00:05:40,13
I have one loose end.
100
00:05:40,13 --> 00:05:44,49
One loose end that I
did not cover yet.
101
00:05:44,49 --> 00:05:49,07
There's one very important
formula that's left, and it's
102
00:05:49,07 --> 00:05:51,92
the derivative of the powers.
103
00:05:51,92 --> 00:05:54,51
We actually didn't do this
- well we did it for
104
00:05:54,51 --> 00:05:57,12
rational numbers r.
105
00:05:57,12 --> 00:06:00,07
So this is the formula here.
106
00:06:00,07 --> 00:06:06,2
But now we're going to check
this for all real numbers, r.
107
00:06:06,2 --> 00:06:09,2
So including all the
irrational ones as well.
108
00:06:09,2 --> 00:06:14,2
This is also good practice
for using base e and using
109
00:06:14,2 --> 00:06:16,48
logarithmic differentiation.
110
00:06:16,48 --> 00:06:21,63
So let me do this by our two
methods that we can use to
111
00:06:21,63 --> 00:06:26,16
handle exponential
type problems.
112
00:06:26,16 --> 00:06:32,07
So method one was base e.
113
00:06:32,07 --> 00:06:34,78
So if I just rewrite this base
e again, that's just this
114
00:06:34,78 --> 00:06:44,53
formula over here. x ^ r =
(e^ln x)^r, which
115
00:06:44,53 --> 00:06:50,35
is e ^ r ln x.
116
00:06:50,35 --> 00:06:55,32
Okay, so now I can
differentiate this.
117
00:06:55,32 --> 00:07:04,65
So I get that d / dx (x^r), now
I'm going to use prime
118
00:07:04,65 --> 00:07:06,86
notation, because I don't want
to keep on writing that d /
119
00:07:06,86 --> 00:07:10,58
dx here; e ^ (r ln x)'.
120
00:07:10,58 --> 00:07:13,89
121
00:07:13,89 --> 00:07:18,61
And now, what I can do is
I can use the chain rule.
122
00:07:18,61 --> 00:07:21,64
The chain rule says that it's
the derivative of a this times
123
00:07:21,64 --> 00:07:24,38
the derivative of the function.
124
00:07:24,38 --> 00:07:29,5
So the derivative of the
exponential is just itself.
125
00:07:29,5 --> 00:07:32,04
And the derivative of this
guy here, well I'll write
126
00:07:32,04 --> 00:07:35,4
it out once, is (r ln x)'.
127
00:07:35,4 --> 00:07:39,7
128
00:07:39,7 --> 00:07:42,25
So what's that equal to?
129
00:07:42,25 --> 00:07:45,89
Well, each of the r ln
x's is just x ^ r.
130
00:07:45,89 --> 00:07:50,13
And this derivative here is...
131
00:07:50,13 --> 00:07:53,87
Well the derivative of r is 0.
132
00:07:53,87 --> 00:07:54,86
This is a constant.
133
00:07:54,86 --> 00:07:56,68
It just factors out.
134
00:07:56,68 --> 00:07:58,94
And ln x now the derivative...
135
00:07:58,94 --> 00:08:01,69
What's the derivative of ln x?
136
00:08:01,69 --> 00:08:06,57
1 / x, so this is going
to be times r / x.
137
00:08:06,57 --> 00:08:10,21
And now, we rewrite it in the
customary form, which is r, we
138
00:08:10,21 --> 00:08:13,04
put the r in front,
x ^ (r - 1).
139
00:08:13,04 --> 00:08:13,82
Ok?
140
00:08:13,82 --> 00:08:19,06
So I just derived the
formula for you.
141
00:08:19,06 --> 00:08:23,31
And it didn't matter whether r
was rational or irrational,
142
00:08:23,31 --> 00:08:25,32
it's the same proof.
143
00:08:25,32 --> 00:08:29,44
Okay so now I have to show you
how method two works as well.
144
00:08:29,44 --> 00:08:35,18
So let's do method two,
which we called logarithmic
145
00:08:35,18 --> 00:08:39,28
differentiation.
146
00:08:39,28 --> 00:08:44,37
And so here I'll use a
symbol, say u for x ^ r, and
147
00:08:44,37 --> 00:08:44,93
I'll take its logarithm.
148
00:08:44,93 --> 00:08:50,48
That's r ln x.
149
00:08:50,48 --> 00:08:51,83
And now I differentiate it.
150
00:08:51,83 --> 00:08:54,99
I'll leave that in the middle,
because I want to remember
151
00:08:54,99 --> 00:08:57,27
the key property of
logarithmic differention.
152
00:08:57,27 --> 00:08:58,95
But first I'll
differentiate it.
153
00:08:58,95 --> 00:09:00,63
Later on, what I'm going
to use is that this
154
00:09:00,63 --> 00:09:02,62
is the same as u'/u.
155
00:09:02,62 --> 00:09:06,47
This is one way of evaluating
a logarithmic derivative.
156
00:09:06,47 --> 00:09:09,23
And then the other is to
differentiate the explicit
157
00:09:09,23 --> 00:09:10,98
function that we
have over here.
158
00:09:10,98 --> 00:09:16,79
And that is just,
as we said, r /x.
159
00:09:16,79 --> 00:09:24,54
So now, I multiply through,
and I get u' = u(r/x), which
160
00:09:24,54 --> 00:09:29,75
is just x^r(r/x), which is
just what we did before.
161
00:09:29,75 --> 00:09:33,41
It's r x ^ (r - 1).
162
00:09:33,41 --> 00:09:38,11
Again, you can now see by
comparing these two pieces
163
00:09:38,11 --> 00:09:41,47
of arithmetic that they're
basically the same.
164
00:09:41,47 --> 00:09:43,95
Pretty much every time you
convert to base c or you do
165
00:09:43,95 --> 00:09:46,09
logarithmic differentiation,
it'll amount to the same
166
00:09:46,09 --> 00:09:48,27
thing, provided you
don't get mixed up.
167
00:09:48,27 --> 00:09:51,72
You generally have to
introduce a new symbol here.
168
00:09:51,72 --> 00:09:55,84
On the other hand, you're
dealing with exponents there.
169
00:09:55,84 --> 00:10:00,99
It's worth it to know
both points of view.
170
00:10:00,99 --> 00:10:05,6
All right, so now I want to
make one last remark before
171
00:10:05,6 --> 00:10:09,91
we finish with exponents.
172
00:10:09,91 --> 00:10:16,34
And, I'll try to sell this to
you in a lot of ways as the
173
00:10:16,34 --> 00:10:21,49
court goes on, but one thing
that I want to try to
174
00:10:21,49 --> 00:10:27,21
emphasize is that the natural
logarithm really is natural.
175
00:10:27,21 --> 00:10:39,92
So, I claim that the
natural log is natural.
176
00:10:39,92 --> 00:10:45,72
And the example that we're
going to use for this
177
00:10:45,72 --> 00:10:53,34
illustration is economics.
178
00:10:53,34 --> 00:10:54,09
Okay?
179
00:10:54,09 --> 00:10:58,52
So let me explain to why the
natural log is the one that's
180
00:10:58,52 --> 00:11:00,82
natural for economics.
181
00:11:00,82 --> 00:11:06,2
If you are imagining the price
of a stock that you own goes
182
00:11:06,2 --> 00:11:11,16
down by a dollar, that's a
totally meaningless statement.
183
00:11:11,16 --> 00:11:13,48
It depends on a lot of things.
184
00:11:13,48 --> 00:11:15,73
In particular, it depends on
whether the original price
185
00:11:15,73 --> 00:11:18,3
was a dollar or 100 dollars.
186
00:11:18,3 --> 00:11:22,13
So there's not much meaning
to these absolute numbers.
187
00:11:22,13 --> 00:11:25,08
It's always the
ratios that matter.
188
00:11:25,08 --> 00:11:30,83
So, for example, I just looked
up an hour ago, the London
189
00:11:30,83 --> 00:11:43,64
Exchange close, and it was down
27.9, which as I said, is
190
00:11:43,64 --> 00:11:47,39
pretty meaningless unless
you know what the actual
191
00:11:47,39 --> 00:11:50,05
total of this index is.
192
00:11:50,05 --> 00:11:54,2
It turns out it was 6,432.
193
00:11:54,2 --> 00:11:59,02
So the change in the price,
divided by the price, which
194
00:11:59,02 --> 00:12:07,55
in this case is 27.9 /
6,432, is what matters.
195
00:12:07,55 --> 00:12:12,01
And, in this case, it
happens to be 0.43%.
196
00:12:12,01 --> 00:12:12,29
All right?
197
00:12:12,29 --> 00:12:14,27
That's what happened today.
198
00:12:14,27 --> 00:12:18,7
And similarly, if you take the
infinitessimal of this, people
199
00:12:18,7 --> 00:12:21,26
think of days as being
relatively small increments
200
00:12:21,26 --> 00:12:25,11
when you're investing in a
stock, you would be interested
201
00:12:25,11 --> 00:12:28,69
in the infinitessimal
sense, p'/p.
202
00:12:28,69 --> 00:12:33,08
The derivative of p / p.
203
00:12:33,08 --> 00:12:35,53
That's just (ln p)'.
204
00:12:35,53 --> 00:12:38,16
205
00:12:38,16 --> 00:12:42,66
So this is the - let me put a
little box around it - the
206
00:12:42,66 --> 00:12:45,46
formula of logarithmic
differentiation.
207
00:12:45,46 --> 00:12:49,7
But let me just emphasize that
it has an actual significance,
208
00:12:49,7 --> 00:12:52,43
and it's the one that's used by
economists and people who
209
00:12:52,43 --> 00:12:54,45
are modeling prices of
things all the time.
210
00:12:54,45 --> 00:12:58,62
They never use absolute prices
when there are large swings.
211
00:12:58,62 --> 00:13:01,01
They always use the
log of the price.
212
00:13:01,01 --> 00:13:07,01
And there's no point in using
log base 10, or log base 2.
213
00:13:07,01 --> 00:13:08,18
Those give you junk.
214
00:13:08,18 --> 00:13:11,19
They give you an extra
factor of log 2.
215
00:13:11,19 --> 00:13:14,87
It's the natural log that's
the obvious one to use.
216
00:13:14,87 --> 00:13:19,38
It's completely straightforward
that this is a simpler
217
00:13:19,38 --> 00:13:22,6
expression than using log base
10 and having a factor of
218
00:13:22,6 --> 00:13:26,8
natural log of 10 there, which
would just mess everything up.
219
00:13:26,8 --> 00:13:29,36
All right, so this is
just one illustration.
220
00:13:29,36 --> 00:13:32,2
Anything that has to do
with ratios is going to
221
00:13:32,2 --> 00:13:36,16
encounter logarithms.
222
00:13:36,16 --> 00:13:41,27
All right, so that's
pretty much it.
223
00:13:41,27 --> 00:13:45,99
That's all I want to
say for now anyway.
224
00:13:45,99 --> 00:13:48,28
There's lots more to say, but
we'll be saying it when we do
225
00:13:48,28 --> 00:13:51,25
applications of derivatives
in the second unit.
226
00:13:51,25 --> 00:13:54,45
So now, what I'd like to
do is to start a review.
227
00:13:54,45 --> 00:13:57,79
I'm just going to run through
what we did in this unit.
228
00:13:57,79 --> 00:14:01,58
I'll tell you approximately
what I expect from you on the
229
00:14:01,58 --> 00:14:06,15
test that's coming up tomorrow.
230
00:14:06,15 --> 00:14:14,64
And, well, so let's get
started with that.
231
00:14:14,64 --> 00:14:27,05
So this is a review
of Unit One.
232
00:14:27,05 --> 00:14:32,61
And I'm just going to put on
the board all of the things
233
00:14:32,61 --> 00:14:35,75
that you need to think about,
anyway, keep in your head.
234
00:14:35,75 --> 00:14:41,75
And there what are
called general formulas
235
00:14:41,75 --> 00:14:45,07
for derivatives.
236
00:14:45,07 --> 00:14:51,97
And then there are
the specific ones.
237
00:14:51,97 --> 00:14:55,92
And let me just remind you what
the general formulas are.
238
00:14:55,92 --> 00:15:01,44
There's what you do to
differentiate a sum, a multiple
239
00:15:01,44 --> 00:15:08,19
of a function, the product
rule, the quotient rule.
240
00:15:08,19 --> 00:15:11,55
Those are several
general formulas.
241
00:15:11,55 --> 00:15:15,02
And then there's one more,
which is the chain rule, which
242
00:15:15,02 --> 00:15:17,45
I'm going to say just a
little bit more about.
243
00:15:17,45 --> 00:15:22,35
So the derivative of a function
of a function is the derivative
244
00:15:22,35 --> 00:15:26,65
of the function times the
derivative of the
245
00:15:26,65 --> 00:15:27,43
other function.
246
00:15:27,43 --> 00:15:33,78
So here, I've
abbreviated u = u(x).
247
00:15:33,78 --> 00:15:36,63
Right, so this is one of
two ways of writing it.
248
00:15:36,63 --> 00:15:40,03
The other way is also one that
you can keep in mind and you
249
00:15:40,03 --> 00:15:42,47
might find easier to remember.
250
00:15:42,47 --> 00:15:46,69
It's probably a good idea
to remember both formulas.
251
00:15:46,69 --> 00:15:53,27
And then the last type of
general formula that we did
252
00:15:53,27 --> 00:15:56,95
was implicit differentiation.
253
00:15:56,95 --> 00:15:59,2
Okay?
254
00:15:59,2 --> 00:16:04,7
So when you do implicit
differentiation, you have an
255
00:16:04,7 --> 00:16:09,27
equation and you don't try to
solve for the unknown function.
256
00:16:09,27 --> 00:16:13,11
You just put it in its simplest
form and you differentiate.
257
00:16:13,11 --> 00:16:20,44
So, we actually did this, in
particular, for inverses.
258
00:16:20,44 --> 00:16:23,805
That was a very, very key
method for calculating the
259
00:16:23,805 --> 00:16:25,18
inverses of functions.
260
00:16:25,18 --> 00:16:28,6
And it's also true that
logarithmic differentiation
261
00:16:28,6 --> 00:16:31,42
is of this type.
262
00:16:31,42 --> 00:16:33,31
This is a transformation.
263
00:16:33,31 --> 00:16:34,84
We're differentiating
something else.
264
00:16:34,84 --> 00:16:37,92
We're transforming the equation
by taking its logarithm
265
00:16:37,92 --> 00:16:40,98
and then differentiating.
266
00:16:40,98 --> 00:16:45,2
Ok, so there are a number of
different ways this is applied.
267
00:16:45,2 --> 00:16:48,45
It can also be applied, anyway,
these are two of them.
268
00:16:48,45 --> 00:16:50,32
So maybe in parenthesis.
269
00:16:50,32 --> 00:16:53,12
These are just examples.
270
00:16:53,12 --> 00:16:54,35
All right.
271
00:16:54,35 --> 00:16:58,55
I'll try to give examples
of at least a few of
272
00:16:58,55 --> 00:16:59,96
these rules later.
273
00:16:59,96 --> 00:17:05,76
So now, the specific functions
that you know how to
274
00:17:05,76 --> 00:17:08,36
differentiate: well you know
how to differentiate now x ^ r
275
00:17:08,36 --> 00:17:11,41
thanks to what I just did.
276
00:17:11,41 --> 00:17:15,77
We have the sine and the cosine
functions, which you're
277
00:17:15,77 --> 00:17:19,5
responsible for knowing what
their derivatives are.
278
00:17:19,5 --> 00:17:26,49
And then other trig functions
like tan and secant.
279
00:17:26,49 --> 00:17:29,81
We generally don't bother with
cosecants and cotangents,
280
00:17:29,81 --> 00:17:32,19
because everything can be
expressed in terms
281
00:17:32,19 --> 00:17:34,03
of these anyway.
282
00:17:34,03 --> 00:17:35,87
Actually, you can really
express everything in terms
283
00:17:35,87 --> 00:17:36,95
of sines and cosines.
284
00:17:36,95 --> 00:17:39,76
But what you'll find is that
it's much more convenient to
285
00:17:39,76 --> 00:17:42,73
remember the derivatives
of these as well.
286
00:17:42,73 --> 00:17:45,87
So memorize all of these.
287
00:17:45,87 --> 00:17:49,81
All right, and then
we had e^x and ln x.
288
00:17:49,81 --> 00:17:53,92
And we had the inverses
of the trig functions.
289
00:17:53,92 --> 00:18:00,01
These were the two that we did:
the arctangent and the arcsin.
290
00:18:00,01 --> 00:18:02,22
So those are the ones
you're responsible for.
291
00:18:02,22 --> 00:18:06,97
You should have enough time,
anyway, to work out anything
292
00:18:06,97 --> 00:18:09,39
else, if you know these.
293
00:18:09,39 --> 00:18:11,8
All right, so basically the
idea is you have a bunch
294
00:18:11,8 --> 00:18:13,07
of special formulas.
295
00:18:13,07 --> 00:18:14,82
You have a bunch of
general formulas.
296
00:18:14,82 --> 00:18:17,87
You put them together,
and you can generate
297
00:18:17,87 --> 00:18:20,97
pretty much anything.
298
00:18:20,97 --> 00:18:24,6
Okay, so let's do a few
examples before going
299
00:18:24,6 --> 00:18:41,29
on with the review.
300
00:18:41,29 --> 00:18:48,23
Okay, so I do want to do a few
examples in sort of increasing
301
00:18:48,23 --> 00:18:50,84
level of difficulty and how you
would combine these
302
00:18:50,84 --> 00:18:51,42
things together.
303
00:18:51,42 --> 00:18:56,38
So first of all, you should
remember that if you
304
00:18:56,38 --> 00:19:02,35
differentiate the secant
function, that's just - oh I
305
00:19:02,35 --> 00:19:04,01
just realized that I wanted
to say something else
306
00:19:04,01 --> 00:19:06,63
before - so forget that.
307
00:19:06,63 --> 00:19:08,06
We'll do that in a second.
308
00:19:08,06 --> 00:19:10,98
I wanted to make some
general remarks.
309
00:19:10,98 --> 00:19:17,29
So there's one rule that you
discussed in my absence,
310
00:19:17,29 --> 00:19:19,07
which is the chain rule.
311
00:19:19,07 --> 00:19:22,1
And I do want to make just a
couple of remarks about the
312
00:19:22,1 --> 00:19:28,1
chain rule now to remind you
of what it is, and also to
313
00:19:28,1 --> 00:19:30,16
present some consequences.
314
00:19:30,16 --> 00:19:39,19
So, a little bit of extra
on the chain rule.
315
00:19:39,19 --> 00:19:43,72
The first thing that I want
say is that we didn't really
316
00:19:43,72 --> 00:19:46,66
fully explain why it's true.
317
00:19:46,66 --> 00:19:54,14
And I do want to just explain
it by example, okay?
318
00:19:54,14 --> 00:20:01,18
So imagine that you have a
function which is, say, 10x
319
00:20:01,18 --> 00:20:02,01
b.
320
00:20:02,01 --> 00:20:02,5
All right?
321
00:20:02,5 --> 00:20:03,69
So y = 10x
322
00:20:03,69 --> 00:20:04,98
b.
323
00:20:04,98 --> 00:20:09,97
Then obviously, y is changing
10 times as fast as b, right?
324
00:20:09,97 --> 00:20:18,06
The issue is this number
here, dy / dx = 10.
325
00:20:18,06 --> 00:20:23,53
And now if x is a function of
something, say t, shifted
326
00:20:23,53 --> 00:20:34,29
by some other constant
here, then dx/dt = 5.
327
00:20:34,29 --> 00:20:38,51
Now all the chain rule is
saying is that if y is going 10
328
00:20:38,51 --> 00:20:44,74
times as fast as t, I'm sorry
as x, and x is going 5 times as
329
00:20:44,74 --> 00:20:50,62
fast as t, then y is going
50 times as fast as t.
330
00:20:50,62 --> 00:20:54,85
And algebraically, all this
means is if I plug in and
331
00:20:54,85 --> 00:20:57,1
substitute, which is what the
composition of the two
332
00:20:57,1 --> 00:21:04,57
functions amounts to, 10(5t +
a) + b and I multiply
333
00:21:04,57 --> 00:21:07,91
it out, I get 50t
334
00:21:07,91 --> 00:21:09,2
10a b.
335
00:21:09,2 --> 00:21:12
Now these terms don't matter.
336
00:21:12 --> 00:21:13,03
The constant terms
don't matter.
337
00:21:13,03 --> 00:21:14,8
The rate is 50.
338
00:21:14,8 --> 00:21:17,6
And so the consequence, if we
put them together, is that
339
00:21:17,6 --> 00:21:30,17
dy/dt = 10*5, which is 50.
340
00:21:30,17 --> 00:21:32,4
All right, so this is
in a nutshell why the
341
00:21:32,4 --> 00:21:33,63
chain rule works.
342
00:21:33,63 --> 00:21:39,45
And why these rates multiply.
343
00:21:39,45 --> 00:21:43,05
The second thing that I wanted
to say about the chain rule is
344
00:21:43,05 --> 00:21:45,89
that it has a few consequences
that make some of the other
345
00:21:45,89 --> 00:21:50,22
rules a little easier to
remember or possibly to avoid.
346
00:21:50,22 --> 00:21:55,46
The messiest rule in my humble
opinion is the quotient
347
00:21:55,46 --> 00:21:59,43
rule, which is kind of
a nuisance to remember.
348
00:21:59,43 --> 00:22:02,49
So let me just remind you, if
you take just the reciprocal
349
00:22:02,49 --> 00:22:05,99
of a function, and you
differentiate it, there's
350
00:22:05,99 --> 00:22:08,17
another way of looking at this.
351
00:22:08,17 --> 00:22:11,03
And it's actually the way that
I use, so I want to encourage
352
00:22:11,03 --> 00:22:12,8
you to think about
it this way too.
353
00:22:12,8 --> 00:22:16,7
This is the same as
(v^-1)' power .
354
00:22:16,7 --> 00:22:18,97
And now instead of using the
quotient rule, which we
355
00:22:18,97 --> 00:22:25,69
could've used, we can use the
chain rule here with the
356
00:22:25,69 --> 00:22:29,72
power -1, which works
by the power law.
357
00:22:29,72 --> 00:22:30,96
So what is this equal to?
358
00:22:30,96 --> 00:22:33,64
This is equal to (-v^-2)(v').
359
00:22:33,64 --> 00:22:38,93
360
00:22:38,93 --> 00:22:42,73
So here, I've applied the
chain rule rather than
361
00:22:42,73 --> 00:22:47,37
the quotient rule.
362
00:22:47,37 --> 00:22:53,74
And similarly, suppose
I wanted to derive the
363
00:22:53,74 --> 00:22:54,56
full quotient rule.
364
00:22:54,56 --> 00:22:57,24
Well, now this may or
may not be easier.
365
00:22:57,24 --> 00:22:59,77
But this is one way of
remembering what's going on.
366
00:22:59,77 --> 00:23:04,67
If you convert it to (uv^-1)
and you differentiate
367
00:23:04,67 --> 00:23:09,18
that, now I can use the
product rule on this.
368
00:23:09,18 --> 00:23:11,43
Of course, I have to
use the chain rule and
369
00:23:11,43 --> 00:23:13,03
this rule as well.
370
00:23:13,03 --> 00:23:15,62
So what do I get?
371
00:23:15,62 --> 00:23:20,69
I get u' , the inverse,
372
00:23:20,69 --> 00:23:22,81
u, and then I have to
differentiate the v inverse.
373
00:23:22,81 --> 00:23:24,49
That's the formula
right up here.
374
00:23:24,49 --> 00:23:25,29
That's (-v^-2)(v').
375
00:23:25,29 --> 00:23:30,3
376
00:23:30,3 --> 00:23:33,23
So that's one way of doing it.
377
00:23:33,23 --> 00:23:35,93
This actually explains the
funny minus sign when you
378
00:23:35,93 --> 00:23:38,56
differentiate v in the formula.
379
00:23:38,56 --> 00:23:41,73
The other formula, the other
way that we did it, was by
380
00:23:41,73 --> 00:23:44,37
putting this over a
common denominator.
381
00:23:44,37 --> 00:23:49,33
The common denominator was v^2.
382
00:23:49,33 --> 00:23:51,58
This comes from this v ^ -2.
383
00:23:51,58 --> 00:23:54,73
And then the second
term is - u v'.
384
00:23:54,73 --> 00:23:57,25
385
00:23:57,25 --> 00:24:00,15
And the first term, we have to
multiply by an extra factor of
386
00:24:00,15 --> 00:24:02,19
v, because we have a v^2
in the denominator.
387
00:24:02,19 --> 00:24:04,32
So it's u'v.
388
00:24:04,32 --> 00:24:08,11
All right, so this is the
quotient rule as we wrote it
389
00:24:08,11 --> 00:24:11,515
down in lecture, and this is
just another way of remembering
390
00:24:11,515 --> 00:24:13,85
it or deriving it without
remembering it, if you just
391
00:24:13,85 --> 00:24:16,7
remember the chain rule
and the product rule.
392
00:24:16,7 --> 00:24:21,46
Okay, so you'll find that in
many contexts, it's easier
393
00:24:21,46 --> 00:24:25,91
to do one or the other.
394
00:24:25,91 --> 00:24:29,21
Okay, so now i'm ready to
differentiate the secant
395
00:24:29,21 --> 00:24:30,99
and a few such functions.
396
00:24:30,99 --> 00:24:36,2
So we'll do some
examples here here.
397
00:24:36,2 --> 00:24:40,42
So here's the secant function,
and I want to use that formula
398
00:24:40,42 --> 00:24:44,82
up there for the reciprocal.
399
00:24:44,82 --> 00:24:48,09
This is the way I think of it.
400
00:24:48,09 --> 00:24:53,15
This is the cos x ^ -1.
401
00:24:53,15 --> 00:24:58,75
And so, the formula
here is just what?
402
00:24:58,75 --> 00:25:04,03
It's just - (cos
x) ^ -2 (-sin x).
403
00:25:04,03 --> 00:25:20,28
404
00:25:20,28 --> 00:25:22,69
So now this is usually written
in a different fashion, so
405
00:25:22,69 --> 00:25:25,17
that's why I'm doing this
for a reason actually.
406
00:25:25,17 --> 00:25:27,81
Which is although there are
several formulas for things,
407
00:25:27,81 --> 00:25:30,51
with trig functions, there
are usually five ways
408
00:25:30,51 --> 00:25:31,87
of writing something.
409
00:25:31,87 --> 00:25:34,43
So I'm writing this one down so
that you know what the standard
410
00:25:34,43 --> 00:25:36,78
way of presenting it is.
411
00:25:36,78 --> 00:25:39,43
So what happens here is
that we have two minus
412
00:25:39,43 --> 00:25:40,3
signs cancelling.
413
00:25:40,3 --> 00:25:44,36
And we get sin x / cos^2 x.
414
00:25:44,36 --> 00:25:48,1
That's a perfectly acceptable
answer, but there's a customary
415
00:25:48,1 --> 00:25:49,47
way in which is written.
416
00:25:49,47 --> 00:25:55,89
It's written (1 / cos
x) (sin x / cos x).
417
00:25:55,89 --> 00:25:58,32
And then we get rid of the
denominators by rewriting
418
00:25:58,32 --> 00:26:04,1
it in terms of secant and
tangent, so sec x tan x.
419
00:26:04,1 --> 00:26:09,63
So this is the form that's
generally used when
420
00:26:09,63 --> 00:26:11,79
you see these formulas
written in textbooks.
421
00:26:11,79 --> 00:26:15,04
And so you know, you need to
watch out, because if you ever
422
00:26:15,04 --> 00:26:18,64
want to use this kind of
Calculus, you'll have not
423
00:26:18,64 --> 00:26:22,84
be put off by all the
secants and tangents.
424
00:26:22,84 --> 00:26:27,27
All right, so getting slightly
more complicated, how about if
425
00:26:27,27 --> 00:26:28,75
we differentiate ln(sec x)?
426
00:26:28,75 --> 00:26:37,4
427
00:26:37,4 --> 00:26:40,48
If you differentiate the
natural log, that's just going
428
00:26:40,48 --> 00:26:49,45
to be (sec x)' / sec x.
429
00:26:49,45 --> 00:26:52,46
And plugging in the formula
that we had before, that's
430
00:26:52,46 --> 00:27:00,33
(sec x) (tan x) / sec
x, which is tan x.
431
00:27:00,33 --> 00:27:03,85
So this one also has
a very nice form.
432
00:27:03,85 --> 00:27:08,23
And you might say that this is
kind of an ugly function, but
433
00:27:08,23 --> 00:27:14,7
the strange thing is that the
natural log was invented before
434
00:27:14,7 --> 00:27:19,56
the exponential by a guy named
Napier, exactly in order to
435
00:27:19,56 --> 00:27:21,72
evaluate functions like this.
436
00:27:21,72 --> 00:27:24,3
These are the functions
that people cared about
437
00:27:24,3 --> 00:27:28,89
a lot, because they were
used in navigation.
438
00:27:28,89 --> 00:27:32,64
You wanted to multiply
sines and cosines together
439
00:27:32,64 --> 00:27:34,03
to do navigation.
440
00:27:34,03 --> 00:27:38,94
And the multiplication he
encoded using a logarithm.
441
00:27:38,94 --> 00:27:40,71
So these were invented
long before people even
442
00:27:40,71 --> 00:27:42,92
knew about exponents.
443
00:27:42,92 --> 00:27:44,6
And it was a surprise,
actually, that it was
444
00:27:44,6 --> 00:27:46,1
connected to exponents.
445
00:27:46,1 --> 00:27:48,74
So the natural log was invented
before the log base 10 and
446
00:27:48,74 --> 00:27:52,65
everything else, exactly
for this kind of purpose.
447
00:27:52,65 --> 00:27:56,51
Anyway, so this is a nice
function, which was very
448
00:27:56,51 --> 00:28:03,77
important, so that your ships
wouldn't crash into the reef.
449
00:28:03,77 --> 00:28:05,57
Okay, let's continue here.
450
00:28:05,57 --> 00:28:09,175
So there's another kind
of function that I want
451
00:28:09,175 --> 00:28:10,49
to discuss with you.
452
00:28:10,49 --> 00:28:16,59
And these are the kinds in
which there's a choice as to
453
00:28:16,59 --> 00:28:19,38
which of these rules to apply.
454
00:28:19,38 --> 00:28:25,13
And I'll just give a couple
of examples of that.
455
00:28:25,13 --> 00:28:28,786
There usually is a better
and a worse way, so let
456
00:28:28,786 --> 00:28:38,43
me illustrate that.
457
00:28:38,43 --> 00:28:41,12
Okay, yet another example.
458
00:28:41,12 --> 00:28:43,83
I hope you've seen
some of these before.
459
00:28:43,83 --> 00:28:45,58
Say (x ^ 10
460
00:28:45,58 --> 00:28:47,28
+ 8x) ^ 6.
461
00:28:47,28 --> 00:28:51,01
462
00:28:51,01 --> 00:28:53,28
So it's a little bit more
complicated than what we had
463
00:28:53,28 --> 00:29:00,33
before, because there were
several more symbols here.
464
00:29:00,33 --> 00:29:03,21
So what should we
do at this point?
465
00:29:03,21 --> 00:29:06,4
There's one choice which I
claim is a bad idea, and
466
00:29:06,4 --> 00:29:10,81
that is to expand this
out to the 6th power.
467
00:29:10,81 --> 00:29:13,53
That's a bad idea,
because it's very long.
468
00:29:13,53 --> 00:29:15,99
And then your answer
will also be very long.
469
00:29:15,99 --> 00:29:19,62
It will fill the entire
exam paper, for instance.
470
00:29:19,62 --> 00:29:20,02
Yeah?
471
00:29:20,02 --> 00:29:21,38
STUDENT: Can you use
the chain rule?
472
00:29:21,38 --> 00:29:21,97
PROFESSOR: Chain rule.
473
00:29:21,97 --> 00:29:22,58
That's it.
474
00:29:22,58 --> 00:29:23,5
We use the chain rule.
475
00:29:23,5 --> 00:29:26,86
So fortunately, this is
relatively easy using
476
00:29:26,86 --> 00:29:27,62
the chain rule.
477
00:29:27,62 --> 00:29:30,79
We just think of this box
as being the function.
478
00:29:30,79 --> 00:29:35,34
And we take 6 times this guy to
the 5th, times the derivative
479
00:29:35,34 --> 00:29:39,38
of this guy, which is 10x ^ 9
480
00:29:39,38 --> 00:29:41,5
8.
481
00:29:41,5 --> 00:29:43,91
And this is, feeling this
in, it's x^10 + 8x.
482
00:29:43,91 --> 00:29:46,14
And that's it.
483
00:29:46,14 --> 00:29:50,27
That's all you need to do
differentiate things like this.
484
00:29:50,27 --> 00:29:55,14
The chain rule is
very effective.
485
00:29:55,14 --> 00:30:00,34
STUDENT: [INAUDIBLE]
486
00:30:00,34 --> 00:30:01,28
PROFESSOR: That's
a good question.
487
00:30:01,28 --> 00:30:05,83
So I'm not really willing to
answer too many questions about
488
00:30:05,83 --> 00:30:07,5
what's going to be on the exam.
489
00:30:07,5 --> 00:30:10,59
But the question that was just
asked is exactly the kind of
490
00:30:10,59 --> 00:30:13,2
question I'm very
happy to answer.
491
00:30:13,2 --> 00:30:20,09
Ok the question was, in what
form is an acceptable answer?
492
00:30:20,09 --> 00:30:23,56
Now in real life, that is a
really serious question.
493
00:30:23,56 --> 00:30:26,38
When you ask a computer a
question and it gives you
494
00:30:26,38 --> 00:30:31,38
500 million sheets of
printout, it's useless.
495
00:30:31,38 --> 00:30:34,85
And you really care what form
answers are in, and indeed,
496
00:30:34,85 --> 00:30:39
somebody might really care what
this thing to the 6th power is,
497
00:30:39 --> 00:30:42,2
and then you would be forced
to discuss things in terms of
498
00:30:42,2 --> 00:30:46,11
that other functional form.
499
00:30:46,11 --> 00:30:50,41
For the purposes of this
exam, this is okay form.
500
00:30:50,41 --> 00:30:54,49
And, in fact, any correct
form is an okay form.
501
00:30:54,49 --> 00:30:57,77
I recommend strongly that you
not try to simplify things
502
00:30:57,77 --> 00:30:59,7
unless we tell you to.
503
00:30:59,7 --> 00:31:04,86
Sometimes it will be to your
advantage to simplify things.
504
00:31:04,86 --> 00:31:08,01
Sometimes we'll say simplify.
505
00:31:08,01 --> 00:31:10,92
It takes a good deal of
experience to know when
506
00:31:10,92 --> 00:31:13,34
it's really worth it to
simplify expressions.
507
00:31:13,34 --> 00:31:13,62
Yes?
508
00:31:13,62 --> 00:31:19,53
STUDENT: [INAUDIBLE]
509
00:31:19,53 --> 00:31:23,59
PROFESSOR: Right, so
turning to this example.
510
00:31:23,59 --> 00:31:25,52
The question is what
is this derivative?
511
00:31:25,52 --> 00:31:27,24
And here's an answer.
512
00:31:27,24 --> 00:31:29,5
That's the end of the problem.
513
00:31:29,5 --> 00:31:31,81
This is a more customary form.
514
00:31:31,81 --> 00:31:37,16
But this is answer is ok.
515
00:31:37,16 --> 00:31:38,61
Same issue.
516
00:31:38,61 --> 00:31:40,97
That's exactly the point.
517
00:31:40,97 --> 00:31:41,66
Yes?
518
00:31:41,66 --> 00:31:51,46
STUDENT: [INAUDIBLE]
519
00:31:51,46 --> 00:31:59,41
PROFESSOR: The question is, do
you have to show the work?
520
00:31:59,41 --> 00:32:00,24
Do you have to show the work?
521
00:32:00,24 --> 00:32:05,87
Well if I ask you what is d/dx
of sec x, then if you wrote
522
00:32:05,87 --> 00:32:08,39
down this answer or you wrote
down this answer showing no
523
00:32:08,39 --> 00:32:11,2
work, that would be acceptable.
524
00:32:11,2 --> 00:32:17,05
If the question was derive the
formula for this from the
525
00:32:17,05 --> 00:32:19,25
formula for the derivative of
the cosine or something like
526
00:32:19,25 --> 00:32:21,16
that, then it would
not be acceptable.
527
00:32:21,16 --> 00:32:24,34
You'd have to carry
out this arithmetic.
528
00:32:24,34 --> 00:32:29,28
So, in other words, typically
this will come up, for
529
00:32:29,28 --> 00:32:32,29
instance, in various contexts.
530
00:32:32,29 --> 00:32:35,04
You just basically have
to follow directions.
531
00:32:35,04 --> 00:32:35,33
Yes?
532
00:32:35,33 --> 00:32:41,52
STUDENT: [INAUDIBLE]
533
00:32:41,52 --> 00:32:43,68
PROFESSOR: The next question
is, are you expected to be able
534
00:32:43,68 --> 00:32:46,18
to prove what the derivative
of the sine function is?
535
00:32:46,18 --> 00:32:49,58
The short answer
to that is yes.
536
00:32:49,58 --> 00:32:52,51
But I will be getting to that
when I discuss the rest
537
00:32:52,51 --> 00:32:54,24
of the material here.
538
00:32:54,24 --> 00:32:58,43
We're almost there.
539
00:32:58,43 --> 00:33:02,64
Okay, so let me just
finish these examples
540
00:33:02,64 --> 00:33:04,88
with one last one.
541
00:33:04,88 --> 00:33:09,04
And then we'll talk about this
question of things like the
542
00:33:09,04 --> 00:33:12,06
derivative of the sine
function, and deriving it.
543
00:33:12,06 --> 00:33:15,62
So the last example that I'd
like to write down is the one
544
00:33:15,62 --> 00:33:20,75
that I promised you in the
first lecture, namely to
545
00:33:20,75 --> 00:33:26,57
differentiate e ^ x arctan x.
546
00:33:26,57 --> 00:33:28,38
Basically you're supposed to
be able to differentiate
547
00:33:28,38 --> 00:33:29,35
any function.
548
00:33:29,35 --> 00:33:32,39
So this is the one that we
mentioned at the beginning.
549
00:33:32,39 --> 00:33:34,13
So here it is.
550
00:33:34,13 --> 00:33:37,28
Let's do it.
551
00:33:37,28 --> 00:33:38,17
So what is it?
552
00:33:38,17 --> 00:33:46,15
Well, it's just equal to -
I have to differentiate.
553
00:33:46,15 --> 00:33:51,28
I have to use the chain rule -
it's equal to the exponential
554
00:33:51,28 --> 00:33:58,2
times the derivative of
this expression here.
555
00:33:58,2 --> 00:33:59,26
That's the chain rule.
556
00:33:59,26 --> 00:34:01,7
That's the first step.
557
00:34:01,7 --> 00:34:06,44
And now I have to apply
the product rule here.
558
00:34:06,44 --> 00:34:10,82
So I have e ^ x arctan x.
559
00:34:10,82 --> 00:34:16,12
And I differentiate the first
factor, so I get arctan x.
560
00:34:16,12 --> 00:34:18,01
Add to it what happens when
I differentiate the second
561
00:34:18,01 --> 00:34:19,67
factor, leaving alone the x.
562
00:34:19,67 --> 00:34:21,64
So that's x / 1
563
00:34:21,64 --> 00:34:24,31
x^2.
564
00:34:24,31 --> 00:34:26,3
And that's it.
565
00:34:26,3 --> 00:34:28,78
That's the end of the problem.
566
00:34:28,78 --> 00:34:30,59
It wasn't that hard.
567
00:34:30,59 --> 00:34:35,56
Of course, it requires you to
remember all of the rules, and
568
00:34:35,56 --> 00:34:37,3
a lot of formulas
underlying them.
569
00:34:37,3 --> 00:34:39,56
So that's consistent with
what I just told you.
570
00:34:39,56 --> 00:34:42,06
I told you that you
wanted to know this.
571
00:34:42,06 --> 00:34:44,74
I told you that you needed
to know this product rule,
572
00:34:44,74 --> 00:34:50,61
and that you needed to
know the chain rule.
573
00:34:50,61 --> 00:34:53,086
And I guess there was one more
thing, the derivative of e to
574
00:34:53,086 --> 00:34:55,26
the x came into play there.
575
00:34:55,26 --> 00:34:59,17
So of these formulas, we
used four of them in
576
00:34:59,17 --> 00:35:03,81
this one calculation.
577
00:35:03,81 --> 00:35:15,88
Okay, so now what other things
did we talk about in Unit One?
578
00:35:15,88 --> 00:35:25,18
So the main other thing that
we talked about was the
579
00:35:25,18 --> 00:35:33,12
definition of a derivative.
580
00:35:33,12 --> 00:35:42,46
And also there was sort of a
goal which was to get to the
581
00:35:42,46 --> 00:35:51,05
meaning of the derivative.
582
00:35:51,05 --> 00:35:56,75
So these are things - so we had
a couple of ways of looking at
583
00:35:56,75 --> 00:35:59,4
it, or at least a couple
that I'm going to
584
00:35:59,4 --> 00:36:01,78
emphasize right now.
585
00:36:01,78 --> 00:36:06,27
But first, let me remind
you what the formula is.
586
00:36:06,27 --> 00:36:13,9
The derivative is the limit as
delta x goes to 0 of (f(x
587
00:36:13,9 --> 00:36:19,04
delta x) - f(x)) / delta x.
588
00:36:19,04 --> 00:36:22,76
So that's it, and this is
certainly going to be
589
00:36:22,76 --> 00:36:25,64
a central focus here.
590
00:36:25,64 --> 00:36:29,6
And you want to be able to
recognize this formula
591
00:36:29,6 --> 00:36:42,76
in a number of ways.
592
00:36:42,76 --> 00:36:44,52
So, how do we use this?
593
00:36:44,52 --> 00:36:50,56
Well one thing we did was
we calculated a bunch of
594
00:36:50,56 --> 00:36:51,45
these rates of change.
595
00:36:51,45 --> 00:36:53,66
In fact, more or less,
they're the ones which are
596
00:36:53,66 --> 00:36:55,76
written right over here.
597
00:36:55,76 --> 00:36:57,21
This list of functions here.
598
00:36:57,21 --> 00:37:01,47
Now, which ones did we start
out with just straight
599
00:37:01,47 --> 00:37:03,8
from the definition here?
600
00:37:03,8 --> 00:37:04,84
Which of these things?
601
00:37:04,84 --> 00:37:05,83
There were a whole
bunch of them.
602
00:37:05,83 --> 00:37:09,18
So we started out with
a function 1 / x.
603
00:37:09,18 --> 00:37:11,53
We did x^n.
604
00:37:11,53 --> 00:37:14,53
We did sine x.
605
00:37:14,53 --> 00:37:16,88
We did cosine x.
606
00:37:16,88 --> 00:37:18,89
Now there was a little
bit of subtlety with
607
00:37:18,89 --> 00:37:21,11
sine x and cosine x.
608
00:37:21,11 --> 00:37:25,21
We got them using
something else.
609
00:37:25,21 --> 00:37:26,88
We didn't quite get
them all the way.
610
00:37:26,88 --> 00:37:31,79
We got them using
the case x = 0.
611
00:37:31,79 --> 00:37:35,59
We got them from the derivative
at x = 0, we got the formulas
612
00:37:35,59 --> 00:37:37,68
for the derivatives
of sine and cosine.
613
00:37:37,68 --> 00:37:41,48
But that was an argument which
involved plug in sine x
614
00:37:41,48 --> 00:37:44,46
delta x, and running through.
615
00:37:44,46 --> 00:37:45,84
So that's one example.
616
00:37:45,84 --> 00:37:50,63
We also did a ^ x.
617
00:37:50,63 --> 00:37:53,27
And that may be it.
618
00:37:53,27 --> 00:37:58,35
Oh yeah, I think
that's about it.
619
00:37:58,35 --> 00:38:00,75
That may be about it.
620
00:38:00,75 --> 00:38:00,95
No.
621
00:38:00,95 --> 00:38:01,62
It isn't.
622
00:38:01,62 --> 00:38:04,62
Ok, so let me make a connection
here which you probably haven't
623
00:38:04,62 --> 00:38:07,77
yet made, which is that
we did it for (u v)'.
624
00:38:07,77 --> 00:38:10,52
625
00:38:10,52 --> 00:38:15,69
And we also did
it for (u / v)'.
626
00:38:15,69 --> 00:38:18,06
So sorry, I shouldn't write
primes, because that's
627
00:38:18,06 --> 00:38:20,5
not consistent.
628
00:38:20,5 --> 00:38:22,74
I differentiated the product;
I differentiated the
629
00:38:22,74 --> 00:38:28,11
quotient using the
same delta x notation.
630
00:38:28,11 --> 00:38:32,76
I guess I forgot that because I
wasn't there when it happened.
631
00:38:32,76 --> 00:38:36,55
So look, these are the
ones that you do by this.
632
00:38:36,55 --> 00:38:39,31
And, of course, you might have
to reduce them to other things.
633
00:38:39,31 --> 00:38:42,19
These involve using
something else.
634
00:38:42,19 --> 00:38:46,61
This one involves using the
slope of this function at 0,
635
00:38:46,61 --> 00:38:48,6
just the way the sine
and the cosine did.
636
00:38:48,6 --> 00:38:51,25
This one involves the
slopes of the individual
637
00:38:51,25 --> 00:38:53,5
functions, u and v.
638
00:38:53,5 --> 00:38:55,33
And this one also
involves the individual.
639
00:38:55,33 --> 00:38:57,13
So, in other words, it doesn't
get you all the way through to
640
00:38:57,13 --> 00:39:01,17
the end, but it's expressed in
terms of something simpler
641
00:39:01,17 --> 00:39:03,01
in each of these cases.
642
00:39:03,01 --> 00:39:05,84
And I could go on.
643
00:39:05,84 --> 00:39:09,69
We didn't do these in class,
but you're certainly... e ^ x
644
00:39:09,69 --> 00:39:12,17
is a perfectly okay one
on one of the exams.
645
00:39:12,17 --> 00:39:14,97
We ask you for 1 / x^2.
646
00:39:14,97 --> 00:39:16,58
In other words, I'm not
claiming that it's going to
647
00:39:16,58 --> 00:39:18,8
be one on this list, but
it certainly can be
648
00:39:18,8 --> 00:39:19,86
any one of these.
649
00:39:19,86 --> 00:39:21,86
But we're not going to ask you
to go all the way through to
650
00:39:21,86 --> 00:39:26,38
the beginning in
these formulas.
651
00:39:26,38 --> 00:39:28,94
There are also some fundamental
limits that I certainly
652
00:39:28,94 --> 00:39:31,21
want you to know about.
653
00:39:31,21 --> 00:39:34,68
And these you can
derive in reverse.
654
00:39:34,68 --> 00:39:58,88
So I will describe that now.
655
00:39:58,88 --> 00:40:06,91
So let me also emphasize the
following thing: I want to
656
00:40:06,91 --> 00:40:18,59
read this backwards now.
657
00:40:18,59 --> 00:40:21,37
This is the theme from the very
beginning of this lecture.
658
00:40:21,37 --> 00:40:25,6
Namely, if you're given
the function f, you can
659
00:40:25,6 --> 00:40:28,31
figure out its derivative
by its formula here.
660
00:40:28,31 --> 00:40:29,86
That is the formula for
this in terms of what's
661
00:40:29,86 --> 00:40:30,92
on the right hand side.
662
00:40:30,92 --> 00:40:35,41
On the other hand, you can
also use the formula in that
663
00:40:35,41 --> 00:40:47,91
direction, and if you know the
slope of something, you can
664
00:40:47,91 --> 00:40:49,17
figure out what the limit is.
665
00:40:49,17 --> 00:40:54,79
For example, I'll use
the letter x here, even
666
00:40:54,79 --> 00:40:56,04
though it's cheating.
667
00:40:56,04 --> 00:40:59,53
Maybe I'll call it delta x
so it's clearer to you.
668
00:40:59,53 --> 00:41:06,9
Maybe I'll call it u.
669
00:41:06,9 --> 00:41:10,37
Suppose you look at
this limit here.
670
00:41:10,37 --> 00:41:15,25
Well, I claim that you should
recognize that is the
671
00:41:15,25 --> 00:41:19,06
derivative with respect to u
of the function e^u at u =
672
00:41:19,06 --> 00:41:22,66
0, which of course
we know to be 1.
673
00:41:22,66 --> 00:41:25,42
So this is reading this
formula in reverse.
674
00:41:25,42 --> 00:41:28,55
It's recognizing that one of
these limits - let me rewrite
675
00:41:28,55 --> 00:41:35,99
this again here - one of these
so-called difference quotient
676
00:41:35,99 --> 00:41:39,39
limits is a derivative.
677
00:41:39,39 --> 00:41:42,19
And since we know a formula
for that derivative,
678
00:41:42,19 --> 00:41:49,94
we can evaluate it.
679
00:41:49,94 --> 00:41:54,77
And lastly, there's one
other type of thing which
680
00:41:54,77 --> 00:41:57,55
I think you should know.
681
00:41:57,55 --> 00:41:59,78
These are the ones you do
with difference quotients.
682
00:41:59,78 --> 00:42:01,59
There are also other
formulas that you want
683
00:42:01,59 --> 00:42:03
to be able to drive.
684
00:42:03 --> 00:42:20,89
You want to be able to
derive formulas by
685
00:42:20,89 --> 00:42:27,67
implicit differentiation.
686
00:42:27,67 --> 00:42:31,73
In other words, the basic idea
is to take whatever equation
687
00:42:31,73 --> 00:42:37,2
you've got and simplify it as
much as possible, without
688
00:42:37,2 --> 00:42:41,26
insisting that you solve for y.
689
00:42:41,26 --> 00:42:44,18
That's not necessarily the
most appropriate way to
690
00:42:44,18 --> 00:42:45,63
get the rate of change.
691
00:42:45,63 --> 00:42:51,91
The much simpler
formula is sin y = x.
692
00:42:51,91 --> 00:42:59,78
And that one is easier to
differentiate implicitly.
693
00:42:59,78 --> 00:43:02,9
So I should say, do
this kind of thing.
694
00:43:02,9 --> 00:43:05,55
So that's, if you like,
a typical derivation
695
00:43:05,55 --> 00:43:08,39
that you might see.
696
00:43:08,39 --> 00:43:13,07
And then there's one last type
of problem that you'll face,
697
00:43:13,07 --> 00:43:21,59
and it's the other thing
that I claim we discussed.
698
00:43:21,59 --> 00:43:26,58
And it goes all the way
back to the first lecture.
699
00:43:26,58 --> 00:43:34,05
So the last thing that we'll be
talking about is tangent lines.
700
00:43:34,05 --> 00:43:34,2
All right?
701
00:43:34,2 --> 00:43:38,76
The geometric point of
view of a derivative.
702
00:43:38,76 --> 00:43:41,9
And we'll be doing more of
this in next the unit.
703
00:43:41,9 --> 00:43:45,41
So first of all, you'll be
expected to be able to
704
00:43:45,41 --> 00:43:52,38
compute the tangent line.
705
00:43:52,38 --> 00:43:56,4
That's often fairly
straightforward.
706
00:43:56,4 --> 00:44:03,59
And the second thing is to
graph y' , the derivative
707
00:44:03,59 --> 00:44:07,37
of a function.
708
00:44:07,37 --> 00:44:10,44
And the third thing, which I'm
going to throw in here, because
709
00:44:10,44 --> 00:44:14,72
I regard it in a sort of
geometric vein, although it's
710
00:44:14,72 --> 00:44:16,69
got an analytical aspect to it.
711
00:44:16,69 --> 00:44:18,87
So this is a picture.
712
00:44:18,87 --> 00:44:20,71
This is a computation.
713
00:44:20,71 --> 00:44:23,33
And if you combine the
two together, you
714
00:44:23,33 --> 00:44:24,27
get something else.
715
00:44:24,27 --> 00:44:37,87
And so this is to recognize
differentiable functions.
716
00:44:37,87 --> 00:44:40,19
Alright, so how do you do this?
717
00:44:40,19 --> 00:44:43,6
Well, we really only have
one way of doing it.
718
00:44:43,6 --> 00:44:54,58
We're going to check the
left and right tangents.
719
00:44:54,58 --> 00:44:59,45
They must be equal.
720
00:44:59,45 --> 00:45:05
So again, this is a property
that you should be familiar
721
00:45:05 --> 00:45:06,83
with from some of
your exercises.
722
00:45:06,83 --> 00:45:09,9
And the idea is simply, that if
the tangent line exists, it's
723
00:45:09,9 --> 00:45:14,77
the same from the right
and from the left.
724
00:45:14,77 --> 00:45:21,94
Ok, now I'm going to just do
one example here from this sort
725
00:45:21,94 --> 00:45:27,76
of qualitative sketching skill
to give you an example here.
726
00:45:27,76 --> 00:45:30,58
And what I'm going to do is
I'm going to draw a graph
727
00:45:30,58 --> 00:45:34,75
of a function like this.
728
00:45:34,75 --> 00:45:38,58
And what I want to do
underneath is draw the
729
00:45:38,58 --> 00:45:41,6
graph of the derivative.
730
00:45:41,6 --> 00:45:46,6
So this is the function y =
f(x), and here I'm going to
731
00:45:46,6 --> 00:45:56,49
draw the graph of the function
y = f'(x) right underneath it.
732
00:45:56,49 --> 00:46:00,33
So now, let's think about what
it's supposed to look like.
733
00:46:00,33 --> 00:46:06,09
And the one step that you need
to make in order to do this, is
734
00:46:06,09 --> 00:46:08,66
to draw a few tangent lines.
735
00:46:08,66 --> 00:46:13,21
I'm just going to
draw one down here.
736
00:46:13,21 --> 00:46:18,73
And I'm going to
draw one up here.
737
00:46:18,73 --> 00:46:23,78
Now, the tangent lines here -
noticed that the slope of these
738
00:46:23,78 --> 00:46:27,3
tangent lines are all positive.
739
00:46:27,3 --> 00:46:31,8
So everything I draw down
here is going to be
740
00:46:31,8 --> 00:46:33,88
above the x-axis.
741
00:46:33,88 --> 00:46:36,58
Furthermore,, as I go further
to the left, they get
742
00:46:36,58 --> 00:46:37,83
steeper and steeper.
743
00:46:37,83 --> 00:46:39,5
So they're getting
higher and higher.
744
00:46:39,5 --> 00:46:44,02
So the function is
coming down like this.
745
00:46:44,02 --> 00:46:45,35
It starts up there.
746
00:46:45,35 --> 00:46:50,57
Maybe I'll draw it in green to
illustrate the graph here.
747
00:46:50,57 --> 00:46:56,91
So that's this function here.
748
00:46:56,91 --> 00:46:59,75
As we get farther out, it's
getting flatter and flatter.
749
00:46:59,75 --> 00:47:06,27
So it's leveling off, but
above the axis like that.
750
00:47:06,27 --> 00:47:09,52
So one of the things to
emphasize is, you should not
751
00:47:09,52 --> 00:47:12,09
expect the derivative to
look like the function.
752
00:47:12,09 --> 00:47:13,32
It's totally different.
753
00:47:13,32 --> 00:47:17,42
It's keeping track at each
point of its tangent line.
754
00:47:17,42 --> 00:47:19,78
On the other hand, you should
get some kind of physical feel
755
00:47:19,78 --> 00:47:23,67
for it, and we'll be practicing
this more in the next unit.
756
00:47:23,67 --> 00:47:25,5
So let me give you an
example of a function
757
00:47:25,5 --> 00:47:27,8
which does exactly this.
758
00:47:27,8 --> 00:47:33,24
And it's the function y = ln x.
759
00:47:33,24 --> 00:47:38,56
If you differentiate it,
you get y' = 1 / x.
760
00:47:38,56 --> 00:47:44,63
And this plot above is, roughly
speaking, the logarithm.
761
00:47:44,63 --> 00:47:50,23
And this plot underneath
is the function 1 / x.
762
00:47:50,23 --> 00:47:53,23
We still have time
for one question.
763
00:47:53,23 --> 00:47:58,58
And so, fire away.
764
00:47:58,58 --> 00:48:03,67
Yes?
765
00:48:03,67 --> 00:48:04,04
STUDENT: [INAUDIBLE]
766
00:48:04,04 --> 00:48:06,75
PROFESSOR: The question is,
can you show how you derive
767
00:48:06,75 --> 00:48:09,77
the inverse tangent of x.
768
00:48:09,77 --> 00:48:13,35
So that's in a lecture.
769
00:48:13,35 --> 00:48:17,78
I'm happy to do it right
now, but it's going to make
770
00:48:17,78 --> 00:48:20,42
me a whole two minutes.
771
00:48:20,42 --> 00:48:27,56
So, here's how you do
it. y = arctan x.
772
00:48:27,56 --> 00:48:30,77
And now this is hopeless
to differentiate, so I
773
00:48:30,77 --> 00:48:34,72
rewrite it as tan y = x.
774
00:48:34,72 --> 00:48:38,44
And now I have to
differentiate that.
775
00:48:38,44 --> 00:48:43,67
So when I differentiate it, I
get the derivative of tan y
776
00:48:43,67 --> 00:48:46,56
with respect to x, so
with respect to y.
777
00:48:46,56 --> 00:48:47,76
That's (1 / 1
778
00:48:47,76 --> 00:48:48,21
y^2 )y'.
779
00:48:48,21 --> 00:48:51,12
780
00:48:51,12 --> 00:48:52,85
So this is a hard step.
781
00:48:52,85 --> 00:48:53,93
That's the chain rule.
782
00:48:53,93 --> 00:48:55,86
And on the left side I get 1.
783
00:48:55,86 --> 00:48:58,73
So I'm doing this super
fast because we have
784
00:48:58,73 --> 00:49:00,72
thirty seconds left.
785
00:49:00,72 --> 00:49:02,84
But this is the hard
step right here.
786
00:49:02,84 --> 00:49:22,81
And it needs for you to know
that d/dy (tan y) = sec^2 y.
787
00:49:22,81 --> 00:49:24,92
So here's the identity.
788
00:49:24,92 --> 00:49:28,5
So you need have known
this in advance.
789
00:49:28,5 --> 00:49:30,74
And that's the input
into this equation.
790
00:49:30,74 --> 00:49:44,44
So now, what we have is that
y' = 1 / sec^2 y, which is
791
00:49:44,44 --> 00:49:51,38
the same thing as cos^2 y.
792
00:49:51,38 --> 00:49:54,98
Now, the last bit of the
problem is to rewrite
793
00:49:54,98 --> 00:49:57,93
this in terms of x.
794
00:49:57,93 --> 00:50:02,664
And that you have to do
with a right triangle.
795
00:50:02,664 --> 00:50:06,884
If this is x and this is 1,
then the angle is y, because
796
00:50:06,884 --> 00:50:09,416
the tangent of y is x.
797
00:50:09,416 --> 00:50:14,902
So this expresses the fact
that the tangent of y is x.
798
00:50:14,902 --> 00:50:18,489
And then the hypoteneuse
is the square root of 1
799
00:50:18,489 --> 00:50:18,7
x^2.
800
00:50:18,7 --> 00:50:21,654
801
00:50:21,654 --> 00:50:27,14
And so the cosine is
1 divided by that.
802
00:50:27,14 --> 00:50:30,305
So this thing is 1 divided
by the square root of 1
803
00:50:30,305 --> 00:50:36,424
x^2, the quantity squared.
804
00:50:36,424 --> 00:50:40,222
So, and then the last little
bit here, since I'm racing
805
00:50:40,222 --> 00:50:44,02
along, is that it's 1 / 1
+ x^2, squared, which I
806
00:50:44,02 --> 00:50:46,13
incorrectly wrote over here.
807
00:50:46,13 --> 00:50:48,662
OK, so good luck on the text.
808
00:50:48,662 --> 00:50:50,806
See you tomorrow.
809
00:50:50,806 --> 00:50:52,17