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PROFESSOR: Right now,
we're finishing up

9
00:00:25,000 --> 00:00:27,740
with the first
unit, and I'd like

10
00:00:27,740 --> 00:00:31,890
to continue in this
lecture, lecture seven,

11
00:00:31,890 --> 00:00:45,510
with some final remarks
about exponents.

12
00:00:45,510 --> 00:00:49,560
So what I'd like to do
is just review something

13
00:00:49,560 --> 00:00:51,910
that I did quickly
last time, and make

14
00:00:51,910 --> 00:00:53,920
a few philosophical
remarks about it.

15
00:00:53,920 --> 00:00:57,660
I think that the steps involved
were maybe a little tricky,

16
00:00:57,660 --> 00:01:00,680
and so I'd like to go
through it one more time.

17
00:01:00,680 --> 00:01:03,900
Remember, that we were
talking about this number a_k,

18
00:01:03,900 --> 00:01:05,510
which is (1 + 1/k)^k.

19
00:01:08,270 --> 00:01:11,400
And what we showed was
that the limit as k

20
00:01:11,400 --> 00:01:16,920
goes to infinity of a_k was e.

21
00:01:16,920 --> 00:01:19,590
So the first thing
that I'd like to do

22
00:01:19,590 --> 00:01:22,890
is just explain the proof
a little bit more clearly

23
00:01:22,890 --> 00:01:28,440
than I did last time
with fewer symbols,

24
00:01:28,440 --> 00:01:31,830
or at least with this
abbreviation of the symbol

25
00:01:31,830 --> 00:01:35,050
here, to show you
what we actually did.

26
00:01:35,050 --> 00:01:41,580
So I'll just remind you
of what we did last time,

27
00:01:41,580 --> 00:01:45,726
and the first
observation was to check,

28
00:01:45,726 --> 00:01:47,350
rather than the limit
of this function,

29
00:01:47,350 --> 00:01:49,550
but to take the log first.

30
00:01:49,550 --> 00:01:51,640
And this is
typically what's done

31
00:01:51,640 --> 00:01:54,810
when you have an exponential,
when you have an exponent.

32
00:01:54,810 --> 00:01:57,750
And what we found was
that the limit here

33
00:01:57,750 --> 00:02:03,500
was 1 as k goes to infinity.

34
00:02:03,500 --> 00:02:05,160
So last time, this
is what we did.

35
00:02:05,160 --> 00:02:07,970
And I just wanted to
be careful and show you

36
00:02:07,970 --> 00:02:09,830
exactly what the next step is.

37
00:02:09,830 --> 00:02:14,360
If you exponentiate this fact;
you take e to this power,

38
00:02:14,360 --> 00:02:21,230
that's going to tend to
e^1, which is just e.

39
00:02:21,230 --> 00:02:26,570
And then, we just observe
that this is the same as a_k.

40
00:02:26,570 --> 00:02:32,180
So the basic ingredient
here is that e^ln a = a.

41
00:02:32,180 --> 00:02:36,750
That's because the log
function is the inverse

42
00:02:36,750 --> 00:02:38,030
of the exponential function.

43
00:02:38,030 --> 00:02:38,860
Yes, question?

44
00:02:38,860 --> 00:02:54,190
STUDENT: [INAUDIBLE]

45
00:02:54,190 --> 00:02:58,180
PROFESSOR: So the question
was, wouldn't the log of this

46
00:02:58,180 --> 00:03:01,530
be 0 because a_k
is tending to 1.

47
00:03:01,530 --> 00:03:03,830
But a_k isn't tending to 1.

48
00:03:03,830 --> 00:03:06,640
Who said it was?

49
00:03:06,640 --> 00:03:10,040
If you take the logarithm,
which is what we did last time,

50
00:03:10,040 --> 00:03:12,880
logarithm of a_k is
indeed k * ln(1 + 1/k).

51
00:03:17,030 --> 00:03:18,520
That does not tend to 0.

52
00:03:18,520 --> 00:03:22,610
This part of it tends to 0, and
this part tends to infinity.

53
00:03:22,610 --> 00:03:26,460
And they balance each
other, 0 times infinity.

54
00:03:26,460 --> 00:03:28,460
We don't really know yet
from this expression,

55
00:03:28,460 --> 00:03:32,189
in fact we did some cleverness
with limits and derivatives,

56
00:03:32,189 --> 00:03:33,230
to figure out this limit.

57
00:03:33,230 --> 00:03:34,354
It was a very subtle thing.

58
00:03:34,354 --> 00:03:37,560
It turned out to be 1.

59
00:03:37,560 --> 00:03:38,850
All right?

60
00:03:38,850 --> 00:03:40,645
Now, the thing that
I'd like to say

61
00:03:40,645 --> 00:03:43,550
- I'm sorry I'm going
to erase this aside here

62
00:03:43,550 --> 00:03:46,100
- but you need to go
back to your notes

63
00:03:46,100 --> 00:03:48,824
and remember that this
is what we did last time.

64
00:03:48,824 --> 00:03:50,490
Because I want to
have room for the next

65
00:03:50,490 --> 00:03:54,420
comment that I want to make on
this little blackboard here.

66
00:03:54,420 --> 00:03:57,090
What we just derived
was this property here,

67
00:03:57,090 --> 00:04:01,810
but I made a claim
yesterday, and I just

68
00:04:01,810 --> 00:04:04,120
want to emphasize it
again so that we realized

69
00:04:04,120 --> 00:04:07,786
what it is that we're doing.

70
00:04:07,786 --> 00:04:08,910
I looked at this backwards.

71
00:04:08,910 --> 00:04:11,430
One way you can think of this
is we're evaluating this limit

72
00:04:11,430 --> 00:04:13,000
and getting an answer.

73
00:04:13,000 --> 00:04:16,524
But all equalities can
be read both directions.

74
00:04:16,524 --> 00:04:17,940
And we can write
it the other way:

75
00:04:17,940 --> 00:04:25,881
e equals the limit, as k goes
to infinity, of this expression

76
00:04:25,881 --> 00:04:26,380
here.

77
00:04:26,380 --> 00:04:28,640
So that's just the same thing.

78
00:04:28,640 --> 00:04:30,840
And if we read it
backwards, what we're saying

79
00:04:30,840 --> 00:04:35,700
is that this limit
is a formula for e.

80
00:04:35,700 --> 00:04:38,325
So this is very
typical of mathematics.

81
00:04:38,325 --> 00:04:40,700
You want to always reverse
your perspective all the time.

82
00:04:40,700 --> 00:04:43,710
Equations work both
ways, and in this case,

83
00:04:43,710 --> 00:04:46,300
we have two different
things here.

84
00:04:46,300 --> 00:04:49,430
This e was what we
defined as the base,

85
00:04:49,430 --> 00:04:54,461
which when you graph e^x,
you get slope 1 at 0.

86
00:04:54,461 --> 00:04:56,710
And then it turns out to be
equal to this limit, which

87
00:04:56,710 --> 00:04:59,060
we can calculate numerically.

88
00:04:59,060 --> 00:05:02,100
If you do this on your
calculators, you, of course,

89
00:05:02,100 --> 00:05:05,420
will have a way of
programming in this number

90
00:05:05,420 --> 00:05:07,290
and evaluating it for each k.

91
00:05:07,290 --> 00:05:10,880
And you'll have another button
available to evaluate this one.

92
00:05:10,880 --> 00:05:12,432
So another way of
saying it is it

93
00:05:12,432 --> 00:05:14,640
that there's a relationship
between these two things.

94
00:05:14,640 --> 00:05:19,210
And all of calculus is a matter
of getting these relationships.

95
00:05:19,210 --> 00:05:21,860
So we can look at these things
in several different ways.

96
00:05:21,860 --> 00:05:23,350
And indeed, that's
what we're going

97
00:05:23,350 --> 00:05:25,832
to be doing at least
at the end of today

98
00:05:25,832 --> 00:05:27,040
in talking about derivatives.

99
00:05:27,040 --> 00:05:29,480
A lot of times when we
talk about derivatives,

100
00:05:29,480 --> 00:05:32,230
we're trying to look at them
from several perspectives

101
00:05:32,230 --> 00:05:34,780
at once.

102
00:05:34,780 --> 00:05:37,280
Okay, so I have to keep
on going with exponents,

103
00:05:37,280 --> 00:05:40,130
because I have one loose end.

104
00:05:40,130 --> 00:05:44,490
One loose end that
I did not cover yet.

105
00:05:44,490 --> 00:05:48,330
There's one very important
formula that's left,

106
00:05:48,330 --> 00:05:51,920
and it's the derivative
of the powers.

107
00:05:51,920 --> 00:05:54,070
We actually didn't
do this - well we

108
00:05:54,070 --> 00:05:57,120
did it for rational numbers r.

109
00:05:57,120 --> 00:06:00,070
So this is the formula here.

110
00:06:00,070 --> 00:06:06,200
But now we're going to check
this for all real numbers, r.

111
00:06:06,200 --> 00:06:09,200
So including all the
irrational ones as well.

112
00:06:09,200 --> 00:06:13,690
This is also good
practice for using base e

113
00:06:13,690 --> 00:06:16,480
and using logarithmic
differentiation.

114
00:06:16,480 --> 00:06:20,810
So let me do this
by our two methods

115
00:06:20,810 --> 00:06:26,160
that we can use to handle
exponential type problems.

116
00:06:26,160 --> 00:06:32,070
So method one was base e.

117
00:06:32,070 --> 00:06:34,040
So if I just rewrite
this base e again,

118
00:06:34,040 --> 00:06:36,910
that's just this
formula over here.

119
00:06:36,910 --> 00:06:50,350
x^r = (e^ln x)^r,
which is e^r ln x.

120
00:06:50,350 --> 00:06:55,320
Okay, so now I can
differentiate this.

121
00:06:55,320 --> 00:07:04,650
So I get that d/dx (x^r),
now I'm going to use prime

122
00:07:04,650 --> 00:07:07,460
notation, because I don't want
to keep on writing that d/dx

123
00:07:07,460 --> 00:07:10,580
here; (e^(r ln x))'.

124
00:07:13,890 --> 00:07:18,457
And now, what I can do is
I can use the chain rule.

125
00:07:18,457 --> 00:07:20,290
The chain rule says
that it's the derivative

126
00:07:20,290 --> 00:07:24,380
of this times the
derivative of the function.

127
00:07:24,380 --> 00:07:29,500
So the derivative of the
exponential is just itself.

128
00:07:29,500 --> 00:07:31,470
And the derivative
of this guy here,

129
00:07:31,470 --> 00:07:35,400
well I'll write it out
once, is (r ln x)'.

130
00:07:39,700 --> 00:07:42,250
So what's that equal to?

131
00:07:42,250 --> 00:07:45,890
Well, e^(r ln x) is is just x^r.

132
00:07:45,890 --> 00:07:53,870
And this derivative here is--
Well the derivative of r is 0.

133
00:07:53,870 --> 00:07:54,860
This is a constant.

134
00:07:54,860 --> 00:07:56,680
It just factors out.

135
00:07:56,680 --> 00:08:02,070
And ln x now has derivative--
What's the derivative of ln x?

136
00:08:02,070 --> 00:08:06,570
1/x, so this is going
to be times r/x.

137
00:08:06,570 --> 00:08:10,180
And now, we rewrite it in the
customary form, which is r,

138
00:08:10,180 --> 00:08:13,040
we put the r in front, x^(r-1).

139
00:08:13,040 --> 00:08:13,820
Okay?

140
00:08:13,820 --> 00:08:19,060
So I just derived
the formula for you.

141
00:08:19,060 --> 00:08:23,310
And it didn't matter whether
r was rational or irrational,

142
00:08:23,310 --> 00:08:25,320
it's the same proof.

143
00:08:25,320 --> 00:08:29,440
Okay so now I have to show you
how method two works as well.

144
00:08:29,440 --> 00:08:34,550
So let's do method
two, which we call

145
00:08:34,550 --> 00:08:39,280
logarithmic differentiation.

146
00:08:39,280 --> 00:08:43,764
And so here I'll use a
symbol, say u, for x^r,

147
00:08:43,764 --> 00:08:44,930
and I'll take its logarithm.

148
00:08:44,930 --> 00:08:50,480
That's r ln x.

149
00:08:50,480 --> 00:08:51,830
And now I differentiate it.

150
00:08:51,830 --> 00:08:54,160
I'll leave that in
the middle, because I

151
00:08:54,160 --> 00:08:55,979
want to remember
the key property

152
00:08:55,979 --> 00:08:57,270
of logarithmic differentiation.

153
00:08:57,270 --> 00:08:58,747
But first I'll differentiate it.

154
00:08:58,747 --> 00:09:01,080
Later on, what I'm going to
use is that this is the same

155
00:09:01,080 --> 00:09:02,620
as u'/u.

156
00:09:02,620 --> 00:09:06,470
This is one way of evaluating
a logarithmic derivative.

157
00:09:06,470 --> 00:09:08,660
And then the other
is to differentiate

158
00:09:08,660 --> 00:09:10,980
the explicit function
that we have over here.

159
00:09:10,980 --> 00:09:16,790
And that is just,
as we said, r/x.

160
00:09:16,790 --> 00:09:25,470
So now, I multiply through, and
I get u' = ur/x which is just

161
00:09:25,470 --> 00:09:29,750
x^r r/x, which is just
what we did before.

162
00:09:29,750 --> 00:09:30,520
It's r x^(r-1).

163
00:09:33,410 --> 00:09:36,790
Again, you can now
see by comparing

164
00:09:36,790 --> 00:09:40,110
these two pieces of arithmetic
that they're basically

165
00:09:40,110 --> 00:09:41,470
the same.

166
00:09:41,470 --> 00:09:43,510
Pretty much every time
you convert to base e

167
00:09:43,510 --> 00:09:45,170
or you do logarithmic
differentiation,

168
00:09:45,170 --> 00:09:46,836
it'll amount to the
same thing, provided

169
00:09:46,836 --> 00:09:48,270
you don't get mixed up.

170
00:09:48,270 --> 00:09:51,720
You generally have to
introduce a new symbol here.

171
00:09:51,720 --> 00:09:55,840
On the other hand, you're
dealing with exponents there.

172
00:09:55,840 --> 00:10:00,990
It's worth it to know
both points of view.

173
00:10:00,990 --> 00:10:07,470
All right, so now I want to
make one last remark before we

174
00:10:07,470 --> 00:10:09,910
finish with exponents.

175
00:10:09,910 --> 00:10:16,120
And, I'll try to sell this
to you in a lot of ways

176
00:10:16,120 --> 00:10:19,490
as the course goes on,
but one thing that I

177
00:10:19,490 --> 00:10:23,370
want to try to emphasize is that
the natural logarithm really

178
00:10:23,370 --> 00:10:27,210
is natural.

179
00:10:27,210 --> 00:10:39,920
So, I claim that the
natural log is natural.

180
00:10:39,920 --> 00:10:45,900
And the example that we're going
to use for this illustration

181
00:10:45,900 --> 00:10:53,340
is economics.

182
00:10:53,340 --> 00:10:54,090
Okay?

183
00:10:54,090 --> 00:10:58,520
So let me explain to why the
natural log is the one that's

184
00:10:58,520 --> 00:11:00,820
natural for economics.

185
00:11:00,820 --> 00:11:06,040
If you are imagining the
price of a stock that you own

186
00:11:06,040 --> 00:11:11,160
goes down by a dollar, that's a
totally meaningless statement.

187
00:11:11,160 --> 00:11:13,439
It depends on a lot of things.

188
00:11:13,439 --> 00:11:15,730
In particular, it depends on
whether the original price

189
00:11:15,730 --> 00:11:18,300
was a dollar or 100 dollars.

190
00:11:18,300 --> 00:11:22,130
So there's not much meaning
to these absolute numbers.

191
00:11:22,130 --> 00:11:25,080
It's always the
ratios that matter.

192
00:11:25,080 --> 00:11:29,280
So, for example, I just
looked up an hour ago,

193
00:11:29,280 --> 00:11:42,050
the London Exchange closed,
and it was down 27.9,

194
00:11:42,050 --> 00:11:44,480
which as I said, is
pretty meaningless

195
00:11:44,480 --> 00:11:50,050
unless you know what the
actual total of this index is.

196
00:11:50,050 --> 00:11:54,200
It turns out it was 6,432.

197
00:11:54,200 --> 00:11:57,070
So the change in
the price, divided

198
00:11:57,070 --> 00:12:03,980
by the price, which in
this case is 27.9 / 6,432,

199
00:12:03,980 --> 00:12:07,550
is what matters.

200
00:12:07,550 --> 00:12:11,791
And, in this case, it
happens to be .43%.

201
00:12:11,791 --> 00:12:12,290
All right?

202
00:12:12,290 --> 00:12:14,270
That's what happened today.

203
00:12:14,270 --> 00:12:18,410
And similarly, if you take
the infinitesimal of this,

204
00:12:18,410 --> 00:12:21,260
people think of days as being
relatively small increments

205
00:12:21,260 --> 00:12:23,900
when you're
investing in a stock,

206
00:12:23,900 --> 00:12:27,240
you would be interested in
the infinitesimal sense,

207
00:12:27,240 --> 00:12:28,690
you would be interested in p'/p.

208
00:12:28,690 --> 00:12:33,080
The derivative of
p divided by p.

209
00:12:33,080 --> 00:12:35,530
That's just (ln p)'.

210
00:12:38,160 --> 00:12:42,275
So this is the - let me
put a little box around it

211
00:12:42,275 --> 00:12:45,460
- the formula of
logarithmic differentiation.

212
00:12:45,460 --> 00:12:49,700
But let me just emphasize that
it has an actual significance,

213
00:12:49,700 --> 00:12:52,430
and it's the one that's used
by economists and people who

214
00:12:52,430 --> 00:12:54,450
are modeling prices of
things all the time.

215
00:12:54,450 --> 00:12:58,620
They never use absolute prices
when there are large swings.

216
00:12:58,620 --> 00:13:01,010
They always use the
log of the price.

217
00:13:01,010 --> 00:13:07,010
And there's no point in using
log base 10, or log base 2.

218
00:13:07,010 --> 00:13:08,180
Those give you junk.

219
00:13:08,180 --> 00:13:11,190
They give you an
extra factor of log 2.

220
00:13:11,190 --> 00:13:14,870
It's the natural log that's
the obvious one to use.

221
00:13:14,870 --> 00:13:18,000
It's completely
straightforward that this

222
00:13:18,000 --> 00:13:21,010
is a simpler expression
than using log base 10

223
00:13:21,010 --> 00:13:24,030
and having a factor of
natural log of 10 there,

224
00:13:24,030 --> 00:13:26,800
which would just
mess everything up.

225
00:13:26,800 --> 00:13:29,360
All right, so this is
just one illustration.

226
00:13:29,360 --> 00:13:31,680
Anything that has
to do with ratios

227
00:13:31,680 --> 00:13:36,160
is going to
encounter logarithms.

228
00:13:36,160 --> 00:13:41,270
All right, so that's
pretty much it.

229
00:13:41,270 --> 00:13:45,822
That's all I want to
say for now anyway.

230
00:13:45,822 --> 00:13:47,280
There's lots more
to say, but we'll

231
00:13:47,280 --> 00:13:50,459
be saying it when we do
applications of derivatives

232
00:13:50,459 --> 00:13:51,250
in the second unit.

233
00:13:51,250 --> 00:13:54,450
So now, what I'd like to
do is to start a review.

234
00:13:54,450 --> 00:13:57,790
I'm just going to run through
what we did in this unit.

235
00:13:57,790 --> 00:13:59,700
I'll tell you
approximately what I

236
00:13:59,700 --> 00:14:06,150
expect from you on the test
that's coming up tomorrow.

237
00:14:06,150 --> 00:14:14,640
And, well, so let's
get started with that.

238
00:14:14,640 --> 00:14:27,050
So this is a review of Unit One.

239
00:14:27,050 --> 00:14:32,610
And I'm just going to put on
the board all of the things

240
00:14:32,610 --> 00:14:35,750
that you need to think about,
anyway, keep in your head.

241
00:14:35,750 --> 00:14:41,750
And there are what are
called general formulas

242
00:14:41,750 --> 00:14:45,070
for derivatives.

243
00:14:45,070 --> 00:14:51,970
And then there are
the specific ones.

244
00:14:51,970 --> 00:14:55,920
And let me just remind you
what the general formulas are.

245
00:14:55,920 --> 00:14:58,750
There's what you
do to differentiate

246
00:14:58,750 --> 00:15:04,190
a sum, a multiple of a
function, the product

247
00:15:04,190 --> 00:15:08,190
rule, the quotient rule.

248
00:15:08,190 --> 00:15:11,550
Those are several
general formulas.

249
00:15:11,550 --> 00:15:13,080
And then there's
one more, which is

250
00:15:13,080 --> 00:15:15,780
the chain rule, which
I'm going to say just

251
00:15:15,780 --> 00:15:17,450
a little bit more about.

252
00:15:17,450 --> 00:15:21,200
So the derivative of a
function of a function

253
00:15:21,200 --> 00:15:26,380
is the derivative of the
function times the derivative

254
00:15:26,380 --> 00:15:27,430
of the other function.

255
00:15:27,430 --> 00:15:33,780
So here, I've
abbreviated u is u(x).

256
00:15:33,780 --> 00:15:36,630
Right, so this is one of
two ways of writing it.

257
00:15:36,630 --> 00:15:39,850
The other way is also one
that you can keep in mind

258
00:15:39,850 --> 00:15:42,470
and you might find
easier to remember.

259
00:15:42,470 --> 00:15:46,690
It's probably a good idea
to remember both formulas.

260
00:15:46,690 --> 00:15:49,660
And then the last type
of general formula

261
00:15:49,660 --> 00:15:56,950
that we did was implicit
differentiation.

262
00:15:56,950 --> 00:15:59,200
Okay?

263
00:15:59,200 --> 00:16:03,190
So when you do implicit
differentiation,

264
00:16:03,190 --> 00:16:06,530
you have an equation
and you don't

265
00:16:06,530 --> 00:16:09,270
try to solve for the
unknown function.

266
00:16:09,270 --> 00:16:13,110
You just put it in its simplest
form and you differentiate.

267
00:16:13,110 --> 00:16:20,440
So, we actually did this,
in particular, for inverses.

268
00:16:20,440 --> 00:16:23,520
That was a very, very key
method for calculating

269
00:16:23,520 --> 00:16:25,180
the inverses of functions.

270
00:16:25,180 --> 00:16:28,600
And it's also true that
logarithmic differentiation

271
00:16:28,600 --> 00:16:31,420
is of this type.

272
00:16:31,420 --> 00:16:33,299
This is a transformation.

273
00:16:33,299 --> 00:16:34,840
We're differentiating
something else.

274
00:16:34,840 --> 00:16:37,920
We're transforming the equation
by taking its logarithm

275
00:16:37,920 --> 00:16:40,980
and then differentiating.

276
00:16:40,980 --> 00:16:45,200
Okay, so there are a number of
different ways this is applied.

277
00:16:45,200 --> 00:16:48,450
It can also be applied,
anyway, these are two of them.

278
00:16:48,450 --> 00:16:50,320
So maybe in parenthesis.

279
00:16:50,320 --> 00:16:53,120
These are just examples.

280
00:16:53,120 --> 00:16:54,350
All right.

281
00:16:54,350 --> 00:16:59,461
I'll try to give examples of
at least a few of these rules

282
00:16:59,461 --> 00:16:59,960
later.

283
00:16:59,960 --> 00:17:05,670
So now, the specific
functions that you know how

284
00:17:05,670 --> 00:17:08,360
to differentiate: well you know
how to differentiate now x^r

285
00:17:08,360 --> 00:17:11,410
thanks to what I just did.

286
00:17:11,410 --> 00:17:15,030
We have the sine and
the cosine functions,

287
00:17:15,030 --> 00:17:17,910
which you're
responsible for knowing

288
00:17:17,910 --> 00:17:19,500
what their derivatives are.

289
00:17:19,500 --> 00:17:26,490
And then other trig functions
like tan and secant.

290
00:17:26,490 --> 00:17:29,810
We generally don't bother
with cosecants and cotangents,

291
00:17:29,810 --> 00:17:32,710
because everything can be
expressed in terms of these

292
00:17:32,710 --> 00:17:33,759
anyway.

293
00:17:33,759 --> 00:17:35,550
Actually, you can really
express everything

294
00:17:35,550 --> 00:17:36,906
in terms of sines and cosines.

295
00:17:36,906 --> 00:17:38,280
But what you'll
find is that it's

296
00:17:38,280 --> 00:17:41,660
much more convenient to remember
the derivatives of these

297
00:17:41,660 --> 00:17:42,730
as well.

298
00:17:42,730 --> 00:17:45,870
So memorize all of these.

299
00:17:45,870 --> 00:17:49,810
All right, and then
we had e^x and ln x.

300
00:17:49,810 --> 00:17:53,920
And we had the inverses
of the trig functions.

301
00:17:53,920 --> 00:18:00,010
These were the two that we did:
the arctangent and the arcsine.

302
00:18:00,010 --> 00:18:02,220
So those are the ones
you're responsible for.

303
00:18:02,220 --> 00:18:06,970
You should have enough time,
anyway, to work out anything

304
00:18:06,970 --> 00:18:09,390
else, if you know these.

305
00:18:09,390 --> 00:18:11,210
All right, so
basically the idea is

306
00:18:11,210 --> 00:18:13,070
you have a bunch of
special formulas.

307
00:18:13,070 --> 00:18:14,820
You have a bunch of
general formulas.

308
00:18:14,820 --> 00:18:16,620
You put them
together, and you can

309
00:18:16,620 --> 00:18:20,970
generate pretty much anything.

310
00:18:20,970 --> 00:18:24,810
Okay, so let's do a few
examples before going on

311
00:18:24,810 --> 00:18:41,290
with the review.

312
00:18:41,290 --> 00:18:48,230
Okay, so I do want to do a few
examples in sort of increasing

313
00:18:48,230 --> 00:18:50,170
level of difficulty
in how you would

314
00:18:50,170 --> 00:18:51,420
combine these things together.

315
00:18:51,420 --> 00:18:55,980
So first of all,
you should remember

316
00:18:55,980 --> 00:19:00,885
that if you differentiate the
secant function, that's just

317
00:19:00,885 --> 00:19:03,780
- oh I just realized that
I wanted to say something

318
00:19:03,780 --> 00:19:06,630
else before - so forget that.

319
00:19:06,630 --> 00:19:08,060
We'll do that in a second.

320
00:19:08,060 --> 00:19:10,980
I wanted to make
some general remarks.

321
00:19:10,980 --> 00:19:17,600
So there's one rule that you
discussed in my absence, which

322
00:19:17,600 --> 00:19:19,070
is the chain rule.

323
00:19:19,070 --> 00:19:21,780
And I do want to make
just a couple of remarks

324
00:19:21,780 --> 00:19:26,160
about the chain rule now to
remind you of what it is,

325
00:19:26,160 --> 00:19:30,160
and also to present
some consequences.

326
00:19:30,160 --> 00:19:39,190
So, a little bit of
extra on the chain rule.

327
00:19:39,190 --> 00:19:43,720
The first thing that I want
say is that we didn't really

328
00:19:43,720 --> 00:19:46,660
fully explain why it's true.

329
00:19:46,660 --> 00:19:54,140
And I do want to just
explain it by example, okay?

330
00:19:54,140 --> 00:19:59,720
So imagine that you have
a function which is, say,

331
00:19:59,720 --> 00:20:02,000
10x + b.

332
00:20:02,000 --> 00:20:02,500
All right?

333
00:20:02,500 --> 00:20:04,980
So y = 10x + b.

334
00:20:04,980 --> 00:20:09,970
Then obviously, y is changing
10 times as fast as b, right?

335
00:20:09,970 --> 00:20:18,060
The issue is this number
here, dy/dx, is 10.

336
00:20:18,060 --> 00:20:20,460
And now if x is a
function of something,

337
00:20:20,460 --> 00:20:34,290
say t, shifted by some other
constant here, then dx/dt = 5.

338
00:20:34,290 --> 00:20:38,800
Now all the chain rule is saying
is that if y is going 10 times

339
00:20:38,800 --> 00:20:44,610
as fast as t, I'm sorry as
x, and x is going 5 times

340
00:20:44,610 --> 00:20:50,620
as fast as t, then y is
going 50 times as fast as t.

341
00:20:50,620 --> 00:20:53,530
And algebraically, all
this means is if I plug

342
00:20:53,530 --> 00:20:57,100
in and substitute, which is
what the composition of the two

343
00:20:57,100 --> 00:21:04,760
functions amounts to, 10(5t +
a) + b and I multiply it out,

344
00:21:04,760 --> 00:21:09,200
I get 50t + 10a + b.

345
00:21:09,200 --> 00:21:11,697
Now these terms don't matter.

346
00:21:11,697 --> 00:21:13,030
The constant terms don't matter.

347
00:21:13,030 --> 00:21:14,800
The rate is 50.

348
00:21:14,800 --> 00:21:17,130
And so the consequence,
if we put them together,

349
00:21:17,130 --> 00:21:30,170
is that dy/dt =
10*5, which is 50.

350
00:21:30,170 --> 00:21:31,990
All right, so this
is in a nutshell

351
00:21:31,990 --> 00:21:33,630
why the chain rule works.

352
00:21:33,630 --> 00:21:39,450
And why these rates multiply.

353
00:21:39,450 --> 00:21:42,460
The second thing that I wanted
to say about the chain rule

354
00:21:42,460 --> 00:21:45,050
is that it has a few
consequences that

355
00:21:45,050 --> 00:21:47,380
make some of the other
rules a little easier

356
00:21:47,380 --> 00:21:50,220
to remember or
possibly to avoid.

357
00:21:50,220 --> 00:21:54,510
The messiest rule
in my humble opinion

358
00:21:54,510 --> 00:21:59,430
is the quotient rule, which is
kind of a nuisance to remember.

359
00:21:59,430 --> 00:22:01,150
So let me just
remind you, if you

360
00:22:01,150 --> 00:22:03,840
take just the reciprocal
of a function,

361
00:22:03,840 --> 00:22:05,990
and you differentiate
it, there's

362
00:22:05,990 --> 00:22:08,170
another way of looking at this.

363
00:22:08,170 --> 00:22:09,720
And it's actually
the way that I use,

364
00:22:09,720 --> 00:22:12,800
so I want to encourage you to
think about it this way too.

365
00:22:12,800 --> 00:22:15,670
This is the same as (v^(-1))'.

366
00:22:15,670 --> 00:22:16,700
.

367
00:22:16,700 --> 00:22:18,840
And now instead of using
the quotient rule, which

368
00:22:18,840 --> 00:22:23,840
we could've used, we can
use the chain rule here

369
00:22:23,840 --> 00:22:29,720
with the power -1, which
works by the power law.

370
00:22:29,720 --> 00:22:30,960
So what is this equal to?

371
00:22:30,960 --> 00:22:33,640
This is equal to -v^(-2) v'.

372
00:22:38,930 --> 00:22:42,730
So here, I've applied the
chain rule rather than

373
00:22:42,730 --> 00:22:47,370
the quotient rule.

374
00:22:47,370 --> 00:22:54,060
And similarly, suppose I wanted
to derive the full quotient

375
00:22:54,060 --> 00:22:54,560
rule.

376
00:22:54,560 --> 00:22:57,240
Well, now this may
or may not be easier.

377
00:22:57,240 --> 00:22:59,770
But this is one way of
remembering what's going on.

378
00:22:59,770 --> 00:23:05,300
If you convert it to uv^(-1)
and you differentiate that,

379
00:23:05,300 --> 00:23:09,180
now I can use the
product rule on this.

380
00:23:09,180 --> 00:23:11,970
Of course, I have to use
the chain rule and this rule

381
00:23:11,970 --> 00:23:13,030
as well.

382
00:23:13,030 --> 00:23:15,620
So what do I get?

383
00:23:15,620 --> 00:23:21,519
I get u', the inverse,
plus u, and then I have

384
00:23:21,519 --> 00:23:22,810
to differentiate the v inverse.

385
00:23:22,810 --> 00:23:24,490
That's the formula
right up here.

386
00:23:24,490 --> 00:23:25,290
That's -v^(-2) v'.

387
00:23:30,300 --> 00:23:33,230
So that's one way of doing it.

388
00:23:33,230 --> 00:23:35,760
This actually explains
the funny minus sign

389
00:23:35,760 --> 00:23:38,560
when you differentiate
v in the formula.

390
00:23:38,560 --> 00:23:41,280
The other formula, the
other way that we did it,

391
00:23:41,280 --> 00:23:44,370
was by putting this over
a common denominator.

392
00:23:44,370 --> 00:23:49,330
The common denominator was v^2.

393
00:23:49,330 --> 00:23:51,580
This comes from this v v^(-2).

394
00:23:51,580 --> 00:23:54,730
And then the second
term is -uv'.

395
00:23:57,250 --> 00:24:00,020
And the first term, we have
to multiply by an extra factor

396
00:24:00,020 --> 00:24:02,190
of v, because we have a
v^2 in the denominator.

397
00:24:02,190 --> 00:24:07,720
So it's u'v. All right, so
this is the quotient rule as we

398
00:24:07,720 --> 00:24:11,247
wrote it down in lecture,
and this is just another way

399
00:24:11,247 --> 00:24:13,580
of remembering it or deriving
it without remembering it,

400
00:24:13,580 --> 00:24:16,700
if you just remember the chain
rule and the product rule.

401
00:24:16,700 --> 00:24:19,710
Okay, so you'll find
that in many contexts,

402
00:24:19,710 --> 00:24:25,910
it's easier to do
one or the other.

403
00:24:25,910 --> 00:24:29,210
Okay, so now I'm ready to
differentiate the secant

404
00:24:29,210 --> 00:24:30,990
and a few such functions.

405
00:24:30,990 --> 00:24:36,200
So we'll do some
examples here here.

406
00:24:36,200 --> 00:24:39,030
So here's the secant
function, and I

407
00:24:39,030 --> 00:24:44,820
want to use that formula up
there for the reciprocal.

408
00:24:44,820 --> 00:24:48,090
This is the way I think of it.

409
00:24:48,090 --> 00:24:53,150
This is the cosine
function to the power -1.

410
00:24:53,150 --> 00:24:58,750
And so, the formula
here is just what?

411
00:24:58,750 --> 00:25:04,030
It's just -(cos
x)^(-2) times -sin x.

412
00:25:20,280 --> 00:25:22,550
So now this is usually written
in a different fashion,

413
00:25:22,550 --> 00:25:25,170
so that's why I'm doing
this for a reason actually.

414
00:25:25,170 --> 00:25:27,810
Which is although there are
several formulas for things,

415
00:25:27,810 --> 00:25:29,810
with trig functions,
there are usually

416
00:25:29,810 --> 00:25:31,854
five ways of writing something.

417
00:25:31,854 --> 00:25:33,520
So I'm writing this
one down so that you

418
00:25:33,520 --> 00:25:36,780
know what the standard
way of presenting it is.

419
00:25:36,780 --> 00:25:39,760
So what happens here is
that we have two minus signs

420
00:25:39,760 --> 00:25:40,300
cancelling.

421
00:25:40,300 --> 00:25:44,360
And we get sin x / cos^2 x.

422
00:25:44,360 --> 00:25:46,430
That's a perfectly
acceptable answer,

423
00:25:46,430 --> 00:25:49,470
but there's a customary
way in which is written.

424
00:25:49,470 --> 00:25:55,890
It's written (1 / cos
x) (sin x / cos x).

425
00:25:55,890 --> 00:25:57,530
And then we get rid
of the denominators

426
00:25:57,530 --> 00:26:00,710
by rewriting it in terms
of secant and tangent,

427
00:26:00,710 --> 00:26:04,100
so sec x tan x.

428
00:26:04,100 --> 00:26:07,680
So this is the form
that's generally

429
00:26:07,680 --> 00:26:11,790
used when you see these
formulas written in textbooks.

430
00:26:11,790 --> 00:26:14,440
And so you know, you
need to watch out,

431
00:26:14,440 --> 00:26:16,810
because if you ever want to
use this kind of calculus,

432
00:26:16,810 --> 00:26:22,840
you'll have not be put off by
all the secants and tangents.

433
00:26:22,840 --> 00:26:26,830
All right, so getting
slightly more complicated,

434
00:26:26,830 --> 00:26:28,750
how about if we
differentiate ln(sec x)?

435
00:26:37,400 --> 00:26:39,560
If you differentiate
the natural log,

436
00:26:39,560 --> 00:26:49,450
that's just going to
be (sec x)' / sec x.

437
00:26:49,450 --> 00:26:51,250
And plugging in
the formula that we

438
00:26:51,250 --> 00:27:00,330
had before, that's sec x tan
x / sec x, which is tan x.

439
00:27:00,330 --> 00:27:03,850
So this one also has
a very nice form.

440
00:27:03,850 --> 00:27:07,940
And you might say that this
is kind of an ugly function,

441
00:27:07,940 --> 00:27:14,120
but the strange thing is that
the natural log was invented

442
00:27:14,120 --> 00:27:19,030
before the exponential by
a guy named Napier, exactly

443
00:27:19,030 --> 00:27:21,720
in order to evaluate
functions like this.

444
00:27:21,720 --> 00:27:25,930
These are the functions that
people cared about a lot,

445
00:27:25,930 --> 00:27:28,890
because they were
used in navigation.

446
00:27:28,890 --> 00:27:32,640
You wanted to multiply
sines and cosines together

447
00:27:32,640 --> 00:27:34,030
to do navigation.

448
00:27:34,030 --> 00:27:38,794
And the multiplication he
encoded using a logarithm.

449
00:27:38,794 --> 00:27:40,710
So these were invented
long before people even

450
00:27:40,710 --> 00:27:42,794
knew about exponents.

451
00:27:42,794 --> 00:27:44,460
And it was a surprise,
actually, that it

452
00:27:44,460 --> 00:27:46,100
was connected to exponents.

453
00:27:46,100 --> 00:27:48,650
So the natural log was
invented before the log base 10

454
00:27:48,650 --> 00:27:52,650
and everything else, exactly
for this kind of purpose.

455
00:27:52,650 --> 00:27:54,550
Anyway, so this is
a nice function,

456
00:27:54,550 --> 00:27:58,010
which was very important,
so that your ships wouldn't

457
00:27:58,010 --> 00:28:03,770
crash into the reef.

458
00:28:03,770 --> 00:28:05,570
Okay, let's continue here.

459
00:28:05,570 --> 00:28:08,870
So there's another
kind of function

460
00:28:08,870 --> 00:28:10,490
that I want to discuss with you.

461
00:28:10,490 --> 00:28:12,460
And these are the
kinds in which there's

462
00:28:12,460 --> 00:28:19,380
a choice as to which of
these rules to apply.

463
00:28:19,380 --> 00:28:25,130
And I'll just give a
couple of examples of that.

464
00:28:25,130 --> 00:28:27,620
There usually is a
better and a worse way,

465
00:28:27,620 --> 00:28:38,430
so let me illustrate that.

466
00:28:38,430 --> 00:28:41,120
Okay, yet another example.

467
00:28:41,120 --> 00:28:43,830
I hope you've seen
some of these before.

468
00:28:43,830 --> 00:28:46,660
Say (x^10 + 8x)^6.

469
00:28:51,010 --> 00:28:52,990
So it's a little bit more
complicated than what

470
00:28:52,990 --> 00:29:00,330
we had before, because there
were several more symbols here.

471
00:29:00,330 --> 00:29:03,210
So what should we
do at this point?

472
00:29:03,210 --> 00:29:06,210
There's one choice which
I claim is a bad idea,

473
00:29:06,210 --> 00:29:10,810
and that is to expand
this out to the 6th power.

474
00:29:10,810 --> 00:29:13,530
That's a bad idea,
because it's very long.

475
00:29:13,530 --> 00:29:15,990
And then your answer
will also be very long.

476
00:29:15,990 --> 00:29:19,521
It will fill the entire
exam paper, for instance.

477
00:29:19,521 --> 00:29:20,020
Yeah?

478
00:29:20,020 --> 00:29:21,380
STUDENT: Can you
use the chain rule?

479
00:29:21,380 --> 00:29:21,970
PROFESSOR: Chain rule.

480
00:29:21,970 --> 00:29:22,580
That's it.

481
00:29:22,580 --> 00:29:23,500
We use the chain rule.

482
00:29:23,500 --> 00:29:26,620
So fortunately, this
is relatively easy

483
00:29:26,620 --> 00:29:27,620
using the chain rule.

484
00:29:27,620 --> 00:29:30,790
We just think of this box
as being the function.

485
00:29:30,790 --> 00:29:34,560
And we take 6 times
this guy to the 5th,

486
00:29:34,560 --> 00:29:37,570
times the derivative of this
guy, which is 10x^9 + 8.

487
00:29:41,500 --> 00:29:43,910
And this is, filling
this in, it's x^10 + 8x.

488
00:29:43,910 --> 00:29:46,140
And that's it.

489
00:29:46,140 --> 00:29:50,270
That's all you need to do
differentiate things like this.

490
00:29:50,270 --> 00:29:55,140
The chain rule is
very effective.

491
00:29:55,140 --> 00:29:59,864
STUDENT: [INAUDIBLE]

492
00:29:59,864 --> 00:30:01,280
PROFESSOR: That's
a good question.

493
00:30:01,280 --> 00:30:04,330
So I'm not really willing
to answer too many questions

494
00:30:04,330 --> 00:30:07,500
about what's going
to be on the exam.

495
00:30:07,500 --> 00:30:09,070
But the question
that was just asked

496
00:30:09,070 --> 00:30:13,200
is exactly the kind of question
I'm very happy to answer.

497
00:30:13,200 --> 00:30:18,350
Okay, the question was,
in what form is-- what

498
00:30:18,350 --> 00:30:20,090
form is an acceptable answer?

499
00:30:20,090 --> 00:30:23,560
Now in real life, that is
a really serious question.

500
00:30:23,560 --> 00:30:25,200
When you ask a
computer a question

501
00:30:25,200 --> 00:30:28,930
and it gives you 500
million sheets of printout,

502
00:30:28,930 --> 00:30:31,380
it's useless.

503
00:30:31,380 --> 00:30:33,960
And you really care what
form answers are in,

504
00:30:33,960 --> 00:30:35,730
and indeed, somebody
might really

505
00:30:35,730 --> 00:30:39,000
care what this thing
to the 6th power is,

506
00:30:39,000 --> 00:30:42,090
and then you would be forced
to discuss things in terms

507
00:30:42,090 --> 00:30:46,110
of that other functional form.

508
00:30:46,110 --> 00:30:50,410
For the purposes of this
exam, this is okay form.

509
00:30:50,410 --> 00:30:54,490
And, in fact, any correct
form is an okay form.

510
00:30:54,490 --> 00:30:57,770
I recommend strongly that you
not try to simplify things

511
00:30:57,770 --> 00:30:59,700
unless we tell you to.

512
00:30:59,700 --> 00:31:04,860
Sometimes it will be to your
advantage to simplify things.

513
00:31:04,860 --> 00:31:08,010
Sometimes we'll say simplify.

514
00:31:08,010 --> 00:31:10,390
It takes a good
deal of experience

515
00:31:10,390 --> 00:31:13,121
to know when it's really worth
it to simplify expressions.

516
00:31:13,121 --> 00:31:13,620
Yes?

517
00:31:13,620 --> 00:31:19,530
STUDENT: [INAUDIBLE]

518
00:31:19,530 --> 00:31:23,590
PROFESSOR: Right, so
turning to this example.

519
00:31:23,590 --> 00:31:25,520
The question is what
is this derivative?

520
00:31:25,520 --> 00:31:27,240
And here's an answer.

521
00:31:27,240 --> 00:31:29,500
That's the end of the problem.

522
00:31:29,500 --> 00:31:31,810
This is a more customary form.

523
00:31:31,810 --> 00:31:37,160
But this is answer is okay.

524
00:31:37,160 --> 00:31:38,610
Same issue.

525
00:31:38,610 --> 00:31:40,970
That's exactly the point.

526
00:31:40,970 --> 00:31:41,660
Yes?

527
00:31:41,660 --> 00:31:51,460
STUDENT: [INAUDIBLE]

528
00:31:51,460 --> 00:31:59,032
PROFESSOR: The question is,
do you have to show the work?

529
00:31:59,032 --> 00:32:00,240
Do you have to show the work?

530
00:32:00,240 --> 00:32:04,870
Well if I ask you
what is d/dx of sec x,

531
00:32:04,870 --> 00:32:06,650
then if you wrote
down this answer

532
00:32:06,650 --> 00:32:09,510
or you wrote down this
answer showing no work,

533
00:32:09,510 --> 00:32:11,200
that would be acceptable.

534
00:32:11,200 --> 00:32:15,950
If the question was derive
the formula for this

535
00:32:15,950 --> 00:32:18,650
from the formula for the
derivative of the cosine

536
00:32:18,650 --> 00:32:21,160
or something like that, then
it would not be acceptable.

537
00:32:21,160 --> 00:32:24,340
You'd have to carry
out this arithmetic.

538
00:32:24,340 --> 00:32:28,470
So, in other words,
typically this

539
00:32:28,470 --> 00:32:32,290
will come up, for instance,
in various contexts.

540
00:32:32,290 --> 00:32:34,830
You just basically have
to follow directions.

541
00:32:34,830 --> 00:32:35,330
Yes?

542
00:32:35,330 --> 00:32:41,424
STUDENT: [INAUDIBLE]

543
00:32:41,424 --> 00:32:43,090
PROFESSOR: The next
question is, are you

544
00:32:43,090 --> 00:32:44,465
expected to be
able to prove what

545
00:32:44,465 --> 00:32:46,180
the derivative of
the sine function is?

546
00:32:46,180 --> 00:32:49,580
The short answer to that is yes.

547
00:32:49,580 --> 00:32:51,630
But I will be getting
to that when I discuss

548
00:32:51,630 --> 00:32:54,240
the rest of the material here.

549
00:32:54,240 --> 00:32:58,430
We're almost there.

550
00:32:58,430 --> 00:33:02,640
Okay, so let me just
finish these examples

551
00:33:02,640 --> 00:33:04,880
with one last one.

552
00:33:04,880 --> 00:33:06,880
And then we'll talk
about this question

553
00:33:06,880 --> 00:33:10,630
of things like the derivative
of the sine function,

554
00:33:10,630 --> 00:33:12,060
and deriving it.

555
00:33:12,060 --> 00:33:15,620
So the last example that I'd
like to write down is the one

556
00:33:15,620 --> 00:33:18,940
that I promised you
in the first lecture,

557
00:33:18,940 --> 00:33:26,172
namely to differentiate
e^(x tan^(-1) x).

558
00:33:26,172 --> 00:33:28,380
Basically you're supposed
to be able to differentiate

559
00:33:28,380 --> 00:33:29,350
any function.

560
00:33:29,350 --> 00:33:32,390
So this is the one that we
mentioned at the beginning.

561
00:33:32,390 --> 00:33:34,130
So here it is.

562
00:33:34,130 --> 00:33:37,280
Let's do it.

563
00:33:37,280 --> 00:33:38,170
So what is it?

564
00:33:38,170 --> 00:33:45,742
Well, it's just equal to
- I have to differentiate.

565
00:33:45,742 --> 00:33:47,200
I have to use the
chain rule - it's

566
00:33:47,200 --> 00:33:52,930
equal to the exponential
times the derivative

567
00:33:52,930 --> 00:33:58,200
of this expression here.

568
00:33:58,200 --> 00:33:59,260
That's the chain rule.

569
00:33:59,260 --> 00:34:01,700
That's the first step.

570
00:34:01,700 --> 00:34:06,440
And now I have to apply
the product rule here.

571
00:34:06,440 --> 00:34:10,820
So I have e^(x tan^(-1) x).

572
00:34:10,820 --> 00:34:15,809
And I differentiate the first
factor, so I get tan^(-1) x.

573
00:34:15,809 --> 00:34:17,600
Add to it what happens
when I differentiate

574
00:34:17,600 --> 00:34:19,670
the second factor,
leaving alone the x.

575
00:34:19,670 --> 00:34:21,450
So that's x / (1+x^2).

576
00:34:24,310 --> 00:34:26,300
And that's it.

577
00:34:26,300 --> 00:34:28,780
That's the end of the problem.

578
00:34:28,780 --> 00:34:30,590
It wasn't that hard.

579
00:34:30,590 --> 00:34:35,330
Of course, it requires you
to remember all of the rules,

580
00:34:35,330 --> 00:34:37,300
and a lot of formulas
underlying them.

581
00:34:37,300 --> 00:34:39,560
So that's consistent with
what I just told you.

582
00:34:39,560 --> 00:34:42,060
I told you that you
wanted to know this.

583
00:34:42,060 --> 00:34:44,740
I told you that you needed
to know this product rule,

584
00:34:44,740 --> 00:34:50,419
and that you needed to
know the chain rule.

585
00:34:50,419 --> 00:34:51,960
And I guess there
was one more thing,

586
00:34:51,960 --> 00:34:55,260
the derivative of e^x
came into play there.

587
00:34:55,260 --> 00:34:59,040
So of these formulas,
we used four of them

588
00:34:59,040 --> 00:35:03,810
in this one calculation.

589
00:35:03,810 --> 00:35:15,880
Okay, so now what other things
did we talk about in Unit One?

590
00:35:15,880 --> 00:35:23,590
So the main other thing
that we talked about

591
00:35:23,590 --> 00:35:33,120
was the definition
of a derivative.

592
00:35:33,120 --> 00:35:40,020
And also there
was sort of a goal

593
00:35:40,020 --> 00:35:51,050
which was to get to the
meaning of the derivative.

594
00:35:51,050 --> 00:35:56,520
So these are things - so we
had a couple of ways of looking

595
00:35:56,520 --> 00:35:59,170
at it, or at least
a couple that I'm

596
00:35:59,170 --> 00:36:01,780
going to emphasize right now.

597
00:36:01,780 --> 00:36:06,270
But first, let me remind
you what the formula is.

598
00:36:06,270 --> 00:36:13,900
The derivative is the limit
as delta x goes to 0 of (f(x +

599
00:36:13,900 --> 00:36:19,040
delta x) - f(x)) / delta x.

600
00:36:19,040 --> 00:36:22,430
So that's it, and
this is certainly

601
00:36:22,430 --> 00:36:25,640
going to be a
central focus here.

602
00:36:25,640 --> 00:36:29,600
And you want to be able
to recognize this formula

603
00:36:29,600 --> 00:36:42,760
in a number of ways.

604
00:36:42,760 --> 00:36:44,520
So, how do we use this?

605
00:36:44,520 --> 00:36:48,950
Well one thing we did
was we calculated a bunch

606
00:36:48,950 --> 00:36:51,450
of these rates of change.

607
00:36:51,450 --> 00:36:53,580
In fact, more or less,
they're the ones which

608
00:36:53,580 --> 00:36:55,760
are written right over here.

609
00:36:55,760 --> 00:36:57,210
This list of functions here.

610
00:36:57,210 --> 00:37:01,470
Now, which ones did we
start out with just straight

611
00:37:01,470 --> 00:37:03,800
from the definition here?

612
00:37:03,800 --> 00:37:04,840
Which of these things?

613
00:37:04,840 --> 00:37:06,215
There were a whole
bunch of them.

614
00:37:06,215 --> 00:37:09,180
So we started out
with a function 1/x.

615
00:37:09,180 --> 00:37:11,530
We did x^n.

616
00:37:11,530 --> 00:37:14,530
We did sine x.

617
00:37:14,530 --> 00:37:16,880
We did cosine x.

618
00:37:16,880 --> 00:37:19,205
Now there was a little
bit of subtlety with sine

619
00:37:19,205 --> 00:37:21,110
x and cosine x.

620
00:37:21,110 --> 00:37:25,210
We got them using
something else.

621
00:37:25,210 --> 00:37:26,880
We didn't quite get
them all the way.

622
00:37:26,880 --> 00:37:31,790
We got them using
the case x = 0.

623
00:37:31,790 --> 00:37:34,530
We got them from the
derivative at x = 0,

624
00:37:34,530 --> 00:37:37,680
we got the formulas for the
derivatives of sine and cosine.

625
00:37:37,680 --> 00:37:40,890
But that was an argument
which involved plugging in sin

626
00:37:40,890 --> 00:37:44,460
(x + delta x), and
running through.

627
00:37:44,460 --> 00:37:45,840
So that's one example.

628
00:37:45,840 --> 00:37:50,630
We also did a^x.

629
00:37:50,630 --> 00:37:53,270
And that may be it.

630
00:37:53,270 --> 00:37:58,350
Oh yeah, I think
that's about it.

631
00:37:58,350 --> 00:38:00,450
That may be about it.

632
00:38:00,450 --> 00:38:00,950
No.

633
00:38:00,950 --> 00:38:01,620
It isn't.

634
00:38:01,620 --> 00:38:03,910
Okay, so let me make a
connection here which you

635
00:38:03,910 --> 00:38:07,770
probably haven't yet made, which
is that we did it for (u v)'.

636
00:38:10,520 --> 00:38:15,690
And we also did it for (u / v)'.

637
00:38:15,690 --> 00:38:17,460
So sorry, I shouldn't
write primes,

638
00:38:17,460 --> 00:38:20,500
because that's not consistent
with the claim there.

639
00:38:20,500 --> 00:38:24,900
I differentiated the product;
I differentiated the quotient

640
00:38:24,900 --> 00:38:28,110
using the same delta x notation.

641
00:38:28,110 --> 00:38:32,760
I guess I forgot that because I
wasn't there when it happened.

642
00:38:32,760 --> 00:38:36,550
So look, these are the
ones that you do by this.

643
00:38:36,550 --> 00:38:39,310
And, of course, you might have
to reduce them to other things.

644
00:38:39,310 --> 00:38:42,190
These involve using
something else.

645
00:38:42,190 --> 00:38:46,610
This one involves using the
slope of this function at 0,

646
00:38:46,610 --> 00:38:48,600
just the way the sine
and the cosine did.

647
00:38:48,600 --> 00:38:52,410
This one involves the slopes
of the individual functions, u

648
00:38:52,410 --> 00:38:54,827
and v. And this one also
involves the individual--

649
00:38:54,827 --> 00:38:56,410
So, in other words,
it doesn't get you

650
00:38:56,410 --> 00:38:58,390
all the way through
to the end, but it's

651
00:38:58,390 --> 00:39:03,010
expressed in terms of something
simpler in each of these cases.

652
00:39:03,010 --> 00:39:05,840
And I could go on.

653
00:39:05,840 --> 00:39:09,280
We didn't do these in class,
but you're certainly--

654
00:39:09,280 --> 00:39:12,170
e^x is a perfectly okay
one on one of the exams.

655
00:39:12,170 --> 00:39:14,561
We ask you for 1/x^2.

656
00:39:14,561 --> 00:39:16,310
In other words, I'm
not claiming that it's

657
00:39:16,310 --> 00:39:18,380
going to be one on this
list, but it certainly

658
00:39:18,380 --> 00:39:19,671
can be any one of these.

659
00:39:19,671 --> 00:39:21,170
But we're not going
to ask you to go

660
00:39:21,170 --> 00:39:26,380
all the way through to the
beginning in these formulas.

661
00:39:26,380 --> 00:39:28,940
There are also some fundamental
limits that I certainly

662
00:39:28,940 --> 00:39:31,210
want you to know about.

663
00:39:31,210 --> 00:39:34,680
And these you can
derive in reverse.

664
00:39:34,680 --> 00:39:58,880
So I will describe that now.

665
00:39:58,880 --> 00:40:06,800
So let me also emphasize
the following thing: I want

666
00:40:06,800 --> 00:40:18,590
to read this backwards now.

667
00:40:18,590 --> 00:40:21,370
This is the theme from the
very beginning of this lecture.

668
00:40:21,370 --> 00:40:25,210
Namely, if you're
given the function f,

669
00:40:25,210 --> 00:40:27,811
you can figure out its
derivative by this formula

670
00:40:27,811 --> 00:40:28,310
here.

671
00:40:28,310 --> 00:40:29,860
That is the formula for
this in terms of what's

672
00:40:29,860 --> 00:40:30,920
on the right hand side.

673
00:40:30,920 --> 00:40:34,400
On the other hand,
you can also use

674
00:40:34,400 --> 00:40:45,280
the formula in that
direction, and if you

675
00:40:45,280 --> 00:40:48,310
know the slope of something,
you can figure out

676
00:40:48,310 --> 00:40:49,170
what the limit is.

677
00:40:49,170 --> 00:40:54,570
For example, I'll use
the letter x here,

678
00:40:54,570 --> 00:40:56,040
even though it's cheating.

679
00:40:56,040 --> 00:40:59,530
Maybe I'll call it delta
x so it's clearer to you.

680
00:40:59,530 --> 00:41:06,900
Maybe I'll call it u.

681
00:41:06,900 --> 00:41:10,370
Suppose you look
at this limit here.

682
00:41:10,370 --> 00:41:14,800
Well, I claim that you
should recognize that is

683
00:41:14,800 --> 00:41:19,870
the derivative with respect to
u of the function e^u at u = 0,

684
00:41:19,870 --> 00:41:22,660
which of course we know to be 1.

685
00:41:22,660 --> 00:41:25,420
So this is reading this
formula in reverse.

686
00:41:25,420 --> 00:41:27,940
It's recognizing that
one of these limits -

687
00:41:27,940 --> 00:41:35,160
let me rewrite this again
here - one of these so-called

688
00:41:35,160 --> 00:41:39,390
difference quotient
limits is a derivative.

689
00:41:39,390 --> 00:41:42,190
And since we know a formula
for that derivative,

690
00:41:42,190 --> 00:41:49,940
we can evaluate it.

691
00:41:49,940 --> 00:41:54,150
And lastly, there's
one other type of thing

692
00:41:54,150 --> 00:41:57,550
which I think you should know.

693
00:41:57,550 --> 00:41:59,767
These are the ones you do
with difference quotients.

694
00:41:59,767 --> 00:42:01,350
There are also other
formulas that you

695
00:42:01,350 --> 00:42:03,000
want to be able to derive.

696
00:42:03,000 --> 00:42:19,740
You want to be able
to derive formulas

697
00:42:19,740 --> 00:42:27,670
by implicit differentiation.

698
00:42:27,670 --> 00:42:30,220
In other words,
the basic idea is

699
00:42:30,220 --> 00:42:32,150
to take whatever
equation you've got

700
00:42:32,150 --> 00:42:36,560
and simplify it as
much as possible,

701
00:42:36,560 --> 00:42:41,260
without insisting
that you solve for y.

702
00:42:41,260 --> 00:42:44,080
That's not necessarily
the most appropriate way

703
00:42:44,080 --> 00:42:45,630
to get the rate of change.

704
00:42:45,630 --> 00:42:51,910
The much simpler
formula is sin y = x.

705
00:42:51,910 --> 00:42:59,780
And that one is easier to
differentiate implicitly.

706
00:42:59,780 --> 00:43:02,900
So I should say, do
this kind of thing.

707
00:43:02,900 --> 00:43:05,550
So that's, if you like,
a typical derivation

708
00:43:05,550 --> 00:43:08,390
that you might see.

709
00:43:08,390 --> 00:43:13,070
And then there's one last type
of problem that you'll face,

710
00:43:13,070 --> 00:43:21,590
and it's the other thing
that I claim we discussed.

711
00:43:21,590 --> 00:43:26,580
And it goes all the way
back to the first lecture.

712
00:43:26,580 --> 00:43:33,700
So the last thing that we'll be
talking about is tangent lines.

713
00:43:33,700 --> 00:43:34,200
All right?

714
00:43:34,200 --> 00:43:38,760
The geometric point of
view of a derivative.

715
00:43:38,760 --> 00:43:41,900
And we'll be doing more
of this in next the unit.

716
00:43:41,900 --> 00:43:44,850
So first of all,
you'll be expected

717
00:43:44,850 --> 00:43:52,380
to be able to compute
the tangent line.

718
00:43:52,380 --> 00:43:56,400
That's often fairly
straightforward.

719
00:43:56,400 --> 00:44:03,100
And the second thing
is to graph y' ,

720
00:44:03,100 --> 00:44:07,370
the derivative of a function.

721
00:44:07,370 --> 00:44:09,350
And the third thing,
which I'm going

722
00:44:09,350 --> 00:44:11,550
to throw in here,
because I regard it

723
00:44:11,550 --> 00:44:14,900
in a sort of geometric
vein, although it's got

724
00:44:14,900 --> 00:44:16,690
an analytical aspect to it.

725
00:44:16,690 --> 00:44:18,870
So this is a picture.

726
00:44:18,870 --> 00:44:20,710
This is a computation.

727
00:44:20,710 --> 00:44:23,080
And if you combine
the two together,

728
00:44:23,080 --> 00:44:24,270
you get something else.

729
00:44:24,270 --> 00:44:37,870
And so this is to recognize
differentiable functions.

730
00:44:37,870 --> 00:44:40,190
Alright, so how do you do this?

731
00:44:40,190 --> 00:44:43,600
Well, we really only
have one way of doing it.

732
00:44:43,600 --> 00:44:54,580
We're going to check the
left and right tangents.

733
00:44:54,580 --> 00:44:59,450
They must be equal.

734
00:44:59,450 --> 00:45:04,280
So again, this is
a property that you

735
00:45:04,280 --> 00:45:06,830
should be familiar with
from some of your exercises.

736
00:45:06,830 --> 00:45:09,660
And the idea is simply, that
if the tangent line exists,

737
00:45:09,660 --> 00:45:14,770
it's the same from the
right and from the left.

738
00:45:14,770 --> 00:45:20,200
Okay, now I'm going to
just do one example here

739
00:45:20,200 --> 00:45:25,450
from this sort of
qualitative sketching skill

740
00:45:25,450 --> 00:45:27,289
to give you an example here.

741
00:45:27,289 --> 00:45:28,830
And what I'm going
to do is I'm going

742
00:45:28,830 --> 00:45:34,750
to draw a graph of a
function like this.

743
00:45:34,750 --> 00:45:38,490
And what I want to
do underneath is draw

744
00:45:38,490 --> 00:45:41,600
the graph of the derivative.

745
00:45:41,600 --> 00:45:45,900
So this is the
function y = f(x),

746
00:45:45,900 --> 00:45:48,350
and here I'm going to draw
the graph of the function y =

747
00:45:48,350 --> 00:45:56,490
f'(x) right underneath it.

748
00:45:56,490 --> 00:46:00,330
So now, let's think about what
it's supposed to look like.

749
00:46:00,330 --> 00:46:05,840
And the one step that you need
to make in order to do this,

750
00:46:05,840 --> 00:46:08,660
is to draw a few tangent lines.

751
00:46:08,660 --> 00:46:13,210
I'm just going to
draw one down here.

752
00:46:13,210 --> 00:46:18,730
And I'm going to
draw one up here.

753
00:46:18,730 --> 00:46:22,740
Now, the tangent
lines here - notice

754
00:46:22,740 --> 00:46:27,266
that the slope of these
tangent lines are all positive.

755
00:46:27,266 --> 00:46:28,640
So everything I
draw down here is

756
00:46:28,640 --> 00:46:33,880
going to be above the x-axis.

757
00:46:33,880 --> 00:46:36,190
Furthermore, as I go
further to the left,

758
00:46:36,190 --> 00:46:37,830
they get steeper and steeper.

759
00:46:37,830 --> 00:46:39,500
So they're getting
higher and higher.

760
00:46:39,500 --> 00:46:44,020
So the function is
coming down like this.

761
00:46:44,020 --> 00:46:45,350
It starts up there.

762
00:46:45,350 --> 00:46:50,570
Maybe I'll draw it in green
to illustrate the graph here.

763
00:46:50,570 --> 00:46:56,910
So that's this function here.

764
00:46:56,910 --> 00:46:59,750
As we get farther out, it's
getting flatter and flatter.

765
00:46:59,750 --> 00:47:06,270
So it's leveling off, but
above the axis like that.

766
00:47:06,270 --> 00:47:08,280
So one of the
things to emphasize

767
00:47:08,280 --> 00:47:10,830
is, you should not
expect the derivative

768
00:47:10,830 --> 00:47:12,090
to look like the function.

769
00:47:12,090 --> 00:47:13,320
It's totally different.

770
00:47:13,320 --> 00:47:17,280
It's keeping track at each
point of its tangent line.

771
00:47:17,280 --> 00:47:19,780
On the other hand, you should
get some kind of physical feel

772
00:47:19,780 --> 00:47:23,619
for it, and we'll be practicing
this more in the next unit.

773
00:47:23,619 --> 00:47:25,660
So let me give you an
example of a function which

774
00:47:25,660 --> 00:47:27,800
does exactly this.

775
00:47:27,800 --> 00:47:33,240
And it's the function y = ln x.

776
00:47:33,240 --> 00:47:38,560
If you differentiate
it, you get y' = 1/x.

777
00:47:38,560 --> 00:47:44,630
And this plot above is, roughly
speaking, the logarithm.

778
00:47:44,630 --> 00:47:50,230
And this plot underneath
is the function 1/x.

779
00:47:50,230 --> 00:47:53,230
We still have time
for one question.

780
00:47:53,230 --> 00:47:58,580
And so, fire away.

781
00:47:58,580 --> 00:48:03,207
Yes?

782
00:48:03,207 --> 00:48:04,040
STUDENT: [INAUDIBLE]

783
00:48:04,040 --> 00:48:05,580
PROFESSOR: The
question is, can you

784
00:48:05,580 --> 00:48:09,770
show how you derive the
inverse tangent of x.

785
00:48:09,770 --> 00:48:13,350
So that's in a lecture.

786
00:48:13,350 --> 00:48:17,060
I'm happy to do it right
now, but it's going

787
00:48:17,060 --> 00:48:20,420
to take me a whole two minutes.

788
00:48:20,420 --> 00:48:27,560
So, here's how you do
it. y = tan^(-1) x.

789
00:48:27,560 --> 00:48:30,230
And now this is hopeless
to differentiate,

790
00:48:30,230 --> 00:48:34,720
so I rewrite it as tan y = x.

791
00:48:34,720 --> 00:48:38,440
And now I have to
differentiate that.

792
00:48:38,440 --> 00:48:42,360
So when I
differentiate it, I get

793
00:48:42,360 --> 00:48:44,260
the derivative of
tan y with respect

794
00:48:44,260 --> 00:48:46,560
to x-- with respect to y.

795
00:48:46,560 --> 00:48:48,210
That's 1 / (1 + y^2) times y'.

796
00:48:51,120 --> 00:48:52,850
So this is a hard step.

797
00:48:52,850 --> 00:48:53,930
That's the chain rule.

798
00:48:53,930 --> 00:48:55,860
And on the left side I get 1.

799
00:48:55,860 --> 00:48:57,650
So I'm doing this
super fast because we

800
00:48:57,650 --> 00:49:00,720
have thirty seconds left.

801
00:49:00,720 --> 00:49:02,840
But this is the hard
step right here.

802
00:49:02,840 --> 00:49:04,900
And it needs for you
to know that d/dy tan

803
00:49:04,900 --> 00:49:14,432
y is equal to one over-- Oh,
bad bad bad, secant squared.

804
00:49:14,432 --> 00:49:22,810
I was ahead of myself so fast.

805
00:49:22,810 --> 00:49:24,920
So here's the identity.

806
00:49:24,920 --> 00:49:28,500
So you need have
known this in advance.

807
00:49:28,500 --> 00:49:30,740
And that's the input
into this equation.

808
00:49:30,740 --> 00:49:44,000
So now, what we have is
that y' = 1 / sec^2 y y,

809
00:49:44,000 --> 00:49:51,380
which is the same
thing as cos^2 y.

810
00:49:51,380 --> 00:49:54,170
Now, the last bit
of the problem is

811
00:49:54,170 --> 00:49:57,930
to rewrite this in terms of x.

812
00:49:57,930 --> 00:50:02,664
And that you have to do
with a right triangle.

813
00:50:02,664 --> 00:50:05,618
If this is x and this
is 1, then the angle

814
00:50:05,618 --> 00:50:09,416
is y, because the
tangent of y is x.

815
00:50:09,416 --> 00:50:14,902
So this expresses the fact
that the tangent of y is x.

816
00:50:14,902 --> 00:50:18,700
And then the hypotenuse is
the square root of 1 + x^2.

817
00:50:21,654 --> 00:50:27,140
And so the cosine is
1 divided by that.

818
00:50:27,140 --> 00:50:30,938
So this thing is 1 divided by
the square root of 1 + x^2,

819
00:50:30,938 --> 00:50:36,424
the quantity squared.

820
00:50:36,424 --> 00:50:40,644
So, and then the last little bit
here, since I'm racing along,

821
00:50:40,644 --> 00:50:45,286
is that it's 1 / (1 + x^2),
which I incorrectly wrote over

822
00:50:45,286 --> 00:50:46,130
here.

823
00:50:46,130 --> 00:50:48,662
Okay, so good luck on the test.

824
00:50:48,662 --> 00:50:50,756
See you tomorrow.