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Professor: So, we're ready
to begin the fifth lecture.
9
00:00:24,720 --> 00:00:25,790
I'm glad to be back.
10
00:00:25,790 --> 00:00:33,240
Thank you for entertaining
my colleague, Haynes Miller.
11
00:00:33,240 --> 00:00:35,600
So, today we're
going to continue
12
00:00:35,600 --> 00:00:40,740
where he started, namely what
he talked about was the chain
13
00:00:40,740 --> 00:00:43,690
rule, which is probably
the most powerful technique
14
00:00:43,690 --> 00:00:45,860
for extending the
kinds of functions
15
00:00:45,860 --> 00:00:47,650
that you can differentiate.
16
00:00:47,650 --> 00:00:50,920
And we're going to use the
chain rule in some rather clever
17
00:00:50,920 --> 00:00:54,410
algebraic ways today.
18
00:00:54,410 --> 00:00:57,420
So the topic for
today is what's known
19
00:00:57,420 --> 00:01:10,670
as implicit differentiation.
20
00:01:10,670 --> 00:01:15,040
So implicit differentiation
is a technique
21
00:01:15,040 --> 00:01:17,690
that allows you to differentiate
a lot of functions you didn't
22
00:01:17,690 --> 00:01:20,970
even know how to find before.
23
00:01:20,970 --> 00:01:24,940
And it's a technique -
let's wait for a few people
24
00:01:24,940 --> 00:01:28,260
to sit down here.
25
00:01:28,260 --> 00:01:29,210
Physics, huh?
26
00:01:29,210 --> 00:01:36,419
Okay, more Physics.
27
00:01:36,419 --> 00:01:37,210
Let's take a break.
28
00:01:37,210 --> 00:01:40,980
You can get those after class.
29
00:01:40,980 --> 00:01:46,220
All right, so we're talking
about implicit differentiation,
30
00:01:46,220 --> 00:01:53,760
and I'm going to illustrate
it by several examples.
31
00:01:53,760 --> 00:01:57,610
So this is one of the most
important and basic formulas
32
00:01:57,610 --> 00:01:59,770
that we've already
covered part way.
33
00:01:59,770 --> 00:02:06,950
Namely, the derivative of
x to a power is ax^(a-1).
34
00:02:06,950 --> 00:02:15,780
Now, what we've got so far is
the exponents, 0, plus or minus
35
00:02:15,780 --> 00:02:19,010
1, plus or minus 2, etc.
36
00:02:19,010 --> 00:02:24,440
You did the positive integer
powers in the first lecture,
37
00:02:24,440 --> 00:02:30,710
and then yesterday
Professor Miller
38
00:02:30,710 --> 00:02:32,470
told you about the
negative powers.
39
00:02:32,470 --> 00:02:35,470
So what we're going
to do right now,
40
00:02:35,470 --> 00:02:39,500
today, is we're
going to consider
41
00:02:39,500 --> 00:02:44,420
the exponents which are rational
numbers, ratios of integers.
42
00:02:44,420 --> 00:02:46,540
So a is m/n.
43
00:02:46,540 --> 00:02:53,380
m and n are integers.
44
00:02:53,380 --> 00:02:55,414
All right, so that's
our goal for right now,
45
00:02:55,414 --> 00:02:56,830
and we're going
to use this method
46
00:02:56,830 --> 00:02:58,150
of implicit differentiation.
47
00:02:58,150 --> 00:03:01,010
In particular, it's important
to realize that this
48
00:03:01,010 --> 00:03:03,260
covers the case m = 1.
49
00:03:03,260 --> 00:03:04,890
And those are the nth roots.
50
00:03:04,890 --> 00:03:07,370
So when we take the
one over n power,
51
00:03:07,370 --> 00:03:09,680
we're going to cover
that right now,
52
00:03:09,680 --> 00:03:13,110
along with many other examples.
53
00:03:13,110 --> 00:03:16,360
So this is our first example.
54
00:03:16,360 --> 00:03:17,700
So how do we get started?
55
00:03:17,700 --> 00:03:20,510
Well we just write down a
formula for the function.
56
00:03:20,510 --> 00:03:24,620
The function is y = x^(m/n).
57
00:03:24,620 --> 00:03:26,620
That's what we're
trying to deal with.
58
00:03:26,620 --> 00:03:30,610
And now there's
really only two steps.
59
00:03:30,610 --> 00:03:38,120
The first step is to take this
equation to the nth power,
60
00:03:38,120 --> 00:03:42,900
so write it y^n = x^m.
61
00:03:42,900 --> 00:03:46,170
Alright, so that's just the
same equation re-written.
62
00:03:46,170 --> 00:03:50,180
And now, what we're
going to do is
63
00:03:50,180 --> 00:03:52,170
we're going to differentiate.
64
00:03:52,170 --> 00:04:01,600
So we're going to apply
d/dx to the equation.
65
00:04:01,600 --> 00:04:05,730
Now why is it that we can apply
it to the second equation, not
66
00:04:05,730 --> 00:04:06,670
the first equation?
67
00:04:06,670 --> 00:04:10,220
So maybe I should call these
equation 1 and equation 2.
68
00:04:10,220 --> 00:04:13,150
So, the point is, we can
apply it to equation 2.
69
00:04:13,150 --> 00:04:17,860
Now, the reason is that we
don't know how to differentiate
70
00:04:17,860 --> 00:04:18,860
x^(m/n).
71
00:04:18,860 --> 00:04:21,320
That's something we
just don't know yet.
72
00:04:21,320 --> 00:04:24,630
But we do know how to
differentiate integer powers.
73
00:04:24,630 --> 00:04:29,080
Those are the things that
we took care of before.
74
00:04:29,080 --> 00:04:32,980
So now we're in shape to be
able to do the differentiation.
75
00:04:32,980 --> 00:04:34,900
So I'm going to write
it out explicitly
76
00:04:34,900 --> 00:04:37,930
over here, without
carrying it out just yet.
77
00:04:37,930 --> 00:04:46,460
That's d/dx of
y^n = d/dx of x^m.
78
00:04:46,460 --> 00:04:51,920
And now you see
this expression here
79
00:04:51,920 --> 00:04:55,020
requires us to do something we
couldn't do before yesterday.
80
00:04:55,020 --> 00:04:58,780
Namely, this y is
a function of x.
81
00:04:58,780 --> 00:05:01,610
So we have to apply
the chain rule here.
82
00:05:01,610 --> 00:05:06,710
So this is the same as - this
is by the chain rule now -
83
00:05:06,710 --> 00:05:13,332
d/dy of y^n times dy/dx.
84
00:05:13,332 --> 00:05:15,790
And then, on the right hand
side, we can just carry it out.
85
00:05:15,790 --> 00:05:17,280
We know the formula.
86
00:05:17,280 --> 00:05:18,420
It's mx^(m-1).
87
00:05:21,550 --> 00:05:24,960
Right, now this is our scheme.
88
00:05:24,960 --> 00:05:29,360
And you'll see in a minute
why we win with this.
89
00:05:29,360 --> 00:05:32,390
So, first of all, there
are two factors here.
90
00:05:32,390 --> 00:05:33,870
One of them is unknown.
91
00:05:33,870 --> 00:05:35,870
In fact, it's what
we're looking for.
92
00:05:35,870 --> 00:05:38,570
But the other one is going
to be a known quantity,
93
00:05:38,570 --> 00:05:40,507
because we know how
to differentiate y
94
00:05:40,507 --> 00:05:42,507
to the n with respect to y.
95
00:05:42,507 --> 00:05:44,340
That's the same formula,
although the letter
96
00:05:44,340 --> 00:05:46,330
has been changed.
97
00:05:46,330 --> 00:05:53,070
And so this is the same as -
I'll write it underneath here -
98
00:05:53,070 --> 00:06:07,040
n y^(n-1) dy/dx = m x^(m-1).
99
00:06:07,040 --> 00:06:14,565
Okay, now comes, if you
like, the non-calculus part
100
00:06:14,565 --> 00:06:15,190
of the problem.
101
00:06:15,190 --> 00:06:17,140
Remember the non-calculus
part of the problem
102
00:06:17,140 --> 00:06:20,030
is always the messier
part of the problem.
103
00:06:20,030 --> 00:06:22,210
So we want to figure
out this formula.
104
00:06:22,210 --> 00:06:25,600
This formula, the
answer over here,
105
00:06:25,600 --> 00:06:29,850
which maybe I'll
put in a box now,
106
00:06:29,850 --> 00:06:33,420
has this expressed much more
simply, only in terms of x.
107
00:06:33,420 --> 00:06:36,140
And what we have to do now
is just solve for dy/dx
108
00:06:36,140 --> 00:06:39,730
using algebra, and then solve
all the way in terms of x.
109
00:06:39,730 --> 00:06:41,700
So, first of all,
we solve for dy/dx.
110
00:06:44,230 --> 00:06:47,930
So I do that by dividing the
factor on the left-hand side.
111
00:06:47,930 --> 00:06:52,960
So I get here mx^(m-1)
divided by ny^(n-1).
112
00:06:56,030 --> 00:07:02,020
And now I'm going to plug in--
so I'll write this as m/n.
113
00:07:02,020 --> 00:07:04,540
This is x^(m-1).
114
00:07:04,540 --> 00:07:10,540
Now over here I'm going to put
in for y, x^(m/n) times n-1.
115
00:07:15,690 --> 00:07:18,700
So now we're almost done,
but unfortunately we
116
00:07:18,700 --> 00:07:22,070
have this mess of exponents
that we have to work out.
117
00:07:22,070 --> 00:07:25,060
I'm going to write
it one more time.
118
00:07:25,060 --> 00:07:28,584
So I already recognize
the factor a out front.
119
00:07:28,584 --> 00:07:30,250
That's not going to
be a problem for me,
120
00:07:30,250 --> 00:07:31,980
and that's what I'm
aiming for here.
121
00:07:31,980 --> 00:07:34,700
But now I have to encode
all of these powers,
122
00:07:34,700 --> 00:07:36,390
so let's just write it.
123
00:07:36,390 --> 00:07:41,790
It's m-1, and then it's
minus the quantity (n-1) m/n.
124
00:07:46,270 --> 00:07:50,250
All right, so that's the law of
exponents applied to this ratio
125
00:07:50,250 --> 00:07:50,750
here.
126
00:07:50,750 --> 00:07:58,410
And then we'll do the arithmetic
over here on the next board.
127
00:07:58,410 --> 00:08:08,700
So we have here m - 1
- (n-1) m/n = m - 1.
128
00:08:08,700 --> 00:08:12,460
And if I multiply n
by this, I get -m.
129
00:08:12,460 --> 00:08:15,560
And if the second factor is
minus minus, that's a plus.
130
00:08:15,560 --> 00:08:18,220
And that's +m/n.
131
00:08:18,220 --> 00:08:21,080
Altogether the two m's cancel.
132
00:08:21,080 --> 00:08:23,740
I have here -1 + m/n.
133
00:08:23,740 --> 00:08:27,492
And lo and behold that's
the same thing as a - 1,
134
00:08:27,492 --> 00:08:29,510
just what we wanted.
135
00:08:29,510 --> 00:08:31,900
All right, so this
equals a x^(n-1).
136
00:08:31,900 --> 00:08:39,560
Okay, again just a
bunch of arithmetic.
137
00:08:39,560 --> 00:08:42,530
From this point forward,
from this substitution
138
00:08:42,530 --> 00:08:51,180
on, it's just the
arithmetic of exponents.
139
00:08:51,180 --> 00:08:58,000
All right, so we've done
our first example here.
140
00:08:58,000 --> 00:09:00,590
I want to give you a
couple more examples,
141
00:09:00,590 --> 00:09:04,030
so let's just continue.
142
00:09:04,030 --> 00:09:08,720
The next example I'll
keep relatively simple.
143
00:09:08,720 --> 00:09:15,590
So we have example two, which is
going to be the function x^2 +
144
00:09:15,590 --> 00:09:18,060
y^2 = 1.
145
00:09:18,060 --> 00:09:21,200
Well, that's not
really a function.
146
00:09:21,200 --> 00:09:29,180
It's a way of defining y as
a function of x implicitly.
147
00:09:29,180 --> 00:09:34,700
There's the idea that I could
solve for y if I wanted to.
148
00:09:34,700 --> 00:09:36,490
And indeed let's do that.
149
00:09:36,490 --> 00:09:42,490
So if you solve for y here, what
happens is you get y^2 = 1 -
150
00:09:42,490 --> 00:09:47,380
x^2, and y is equal to plus or
minus the square root of 1 -
151
00:09:47,380 --> 00:09:52,940
x^2.
152
00:09:52,940 --> 00:09:58,200
So this, if you like, is
the implicit definition.
153
00:09:58,200 --> 00:10:00,910
And here is the
explicit function y,
154
00:10:00,910 --> 00:10:04,140
which is a function of x.
155
00:10:04,140 --> 00:10:06,360
And now just for
my own convenience,
156
00:10:06,360 --> 00:10:09,310
I'm just going to take
the positive branch.
157
00:10:09,310 --> 00:10:13,450
This is the function.
158
00:10:13,450 --> 00:10:15,900
It's just really a
circle in disguise.
159
00:10:15,900 --> 00:10:19,600
And I'm just going to take
the top part of the circle,
160
00:10:19,600 --> 00:10:24,920
so we'll take that
top hump here.
161
00:10:24,920 --> 00:10:27,900
All right, so that means
I'm erasing this minus sign.
162
00:10:27,900 --> 00:10:35,660
I'm just taking the
positive branch, just
163
00:10:35,660 --> 00:10:36,470
for my convenience.
164
00:10:36,470 --> 00:10:40,980
I could do it just as well
with the negative branch.
165
00:10:40,980 --> 00:10:46,000
Alright, so now I've
taken the solution,
166
00:10:46,000 --> 00:10:49,540
and I can differentiate
with this.
167
00:10:49,540 --> 00:10:53,070
So rather than using the
dy/dx notation over here,
168
00:10:53,070 --> 00:10:55,510
I'm going to switch
notations over here,
169
00:10:55,510 --> 00:10:56,620
because it's less writing.
170
00:10:56,620 --> 00:10:59,670
I'm going to write y'
and change notations.
171
00:10:59,670 --> 00:11:04,450
Okay, so I want to take
the derivative of this.
172
00:11:04,450 --> 00:11:11,570
Well this is a somewhat
complicated function here.
173
00:11:11,570 --> 00:11:15,770
It's the square root of 1 -
x^2, and the right way always
174
00:11:15,770 --> 00:11:21,720
to look at functions like
this is to rewrite them using
175
00:11:21,720 --> 00:11:26,210
the fractional power notation.
176
00:11:26,210 --> 00:11:28,910
That's the first
step in computing
177
00:11:28,910 --> 00:11:32,600
a derivative of a square root.
178
00:11:32,600 --> 00:11:38,030
And then the second
step here is what?
179
00:11:38,030 --> 00:11:40,740
Does somebody want to tell me?
180
00:11:40,740 --> 00:11:43,650
Chain rule, right.
181
00:11:43,650 --> 00:11:44,300
That's it.
182
00:11:44,300 --> 00:11:45,320
So we have two things.
183
00:11:45,320 --> 00:11:47,720
We start with one, and then
we do something else to it.
184
00:11:47,720 --> 00:11:50,090
So whenever we do two
things to something,
185
00:11:50,090 --> 00:11:52,020
we need to apply the chain rule.
186
00:11:52,020 --> 00:11:55,240
So 1 - x^2, square root.
187
00:11:55,240 --> 00:11:57,180
All right, so how do we do that?
188
00:11:57,180 --> 00:11:58,830
Well, the first
factor I claim is
189
00:11:58,830 --> 00:12:01,220
the derivative of this thing.
190
00:12:01,220 --> 00:12:06,500
So this is 1/2 blah to the -1/2.
191
00:12:06,500 --> 00:12:09,760
So I'm doing this kind
of by the advanced method
192
00:12:09,760 --> 00:12:11,390
now, because we've
already graduated.
193
00:12:11,390 --> 00:12:14,450
You already did the
chain rule last time.
194
00:12:14,450 --> 00:12:15,920
So what does this mean?
195
00:12:15,920 --> 00:12:20,940
This is an abbreviation for
the derivative with respect
196
00:12:20,940 --> 00:12:27,697
to blah of blah ^
1/2, whatever it is.
197
00:12:27,697 --> 00:12:30,280
All right, so that's the first
factor that we're going to use.
198
00:12:30,280 --> 00:12:34,480
Rather than actually write
out a variable for it
199
00:12:34,480 --> 00:12:36,890
and pass through
as I did previously
200
00:12:36,890 --> 00:12:39,160
with this y and x
variable here, I'm
201
00:12:39,160 --> 00:12:41,840
just going to skip
that step and let
202
00:12:41,840 --> 00:12:45,370
you imagine it as being a
placeholder for that variable
203
00:12:45,370 --> 00:12:45,870
here.
204
00:12:45,870 --> 00:12:48,960
So this variable
is now parenthesis.
205
00:12:48,960 --> 00:12:52,370
And then I have to multiply that
by the rate of change of what's
206
00:12:52,370 --> 00:12:55,050
inside with respect to x.
207
00:12:55,050 --> 00:12:58,580
And that is going to be -2x.
208
00:12:58,580 --> 00:13:02,830
The derivative of
1 - x^2 is -2x.
209
00:13:02,830 --> 00:13:09,030
And now again, we couldn't
have done this example two
210
00:13:09,030 --> 00:13:11,730
before example one,
because we needed
211
00:13:11,730 --> 00:13:17,470
to know that the power rule
worked not just for a integer
212
00:13:17,470 --> 00:13:19,690
but also for a = 1/2.
213
00:13:19,690 --> 00:13:22,740
We're using the case
a = 1/2 right here.
214
00:13:22,740 --> 00:13:29,540
It's 1/2 times, and
this -1/2 here is a-1. -
215
00:13:29,540 --> 00:13:33,430
So this is the case a = 1/2.
216
00:13:33,430 --> 00:13:39,790
a-1 happens to be -1/2.
217
00:13:39,790 --> 00:13:41,940
Okay, so I'm putting all
those things together.
218
00:13:41,940 --> 00:13:44,380
And you know within
a week you have
219
00:13:44,380 --> 00:13:45,970
to be doing this
very automatically.
220
00:13:45,970 --> 00:13:47,900
So we're going to do
it at this speed now.
221
00:13:47,900 --> 00:13:49,780
You want to do it even
faster, ultimately.
222
00:13:49,780 --> 00:13:50,280
Yes?
223
00:13:50,280 --> 00:13:53,630
Student: [INAUDIBLE]
224
00:13:53,630 --> 00:13:56,110
Professor: The question is
could I have done it implicitly
225
00:13:56,110 --> 00:13:58,060
without the square roots.
226
00:13:58,060 --> 00:13:59,440
And the answer is yes.
227
00:13:59,440 --> 00:14:02,040
That's what I'm about to do.
228
00:14:02,040 --> 00:14:04,570
So this is an
illustration of what's
229
00:14:04,570 --> 00:14:07,430
called the explicit solution.
230
00:14:07,430 --> 00:14:13,800
So this guy is what's
called explicit.
231
00:14:13,800 --> 00:14:17,024
And I want to contrast
it with the method
232
00:14:17,024 --> 00:14:18,440
that we're going
to now use today.
233
00:14:18,440 --> 00:14:20,410
So it involves a lot
of complications.
234
00:14:20,410 --> 00:14:21,701
It involves the chain rule.
235
00:14:21,701 --> 00:14:23,700
And as we'll see it can
get messier and messier.
236
00:14:23,700 --> 00:14:27,260
And then there's
the implicit method,
237
00:14:27,260 --> 00:14:29,830
which I claim is easier.
238
00:14:29,830 --> 00:14:36,170
So let's see what happens
if you do it implicitly
239
00:14:36,170 --> 00:14:41,010
The implicit method
involves, instead of writing
240
00:14:41,010 --> 00:14:43,610
the function in this
relatively complicated way,
241
00:14:43,610 --> 00:14:47,380
with the square root, it
involves leaving it alone.
242
00:14:47,380 --> 00:14:50,050
Don't do anything to it.
243
00:14:50,050 --> 00:14:52,820
In this previous case, we were
left with something which was
244
00:14:52,820 --> 00:14:56,820
complicated, say x^(1/3)
or x^(1/2) or something
245
00:14:56,820 --> 00:14:57,370
complicated.
246
00:14:57,370 --> 00:14:59,326
We had to simplify it.
247
00:14:59,326 --> 00:15:01,450
We had an equation one,
which was more complicated.
248
00:15:01,450 --> 00:15:03,970
We simplified it then
differentiated it.
249
00:15:03,970 --> 00:15:05,590
And so that was a simpler case.
250
00:15:05,590 --> 00:15:09,590
Well here, the simplest
thing us to differentiate
251
00:15:09,590 --> 00:15:13,470
is the one we started with,
because squares are practically
252
00:15:13,470 --> 00:15:16,725
the easiest thing after first
powers, or maybe zeroth powers
253
00:15:16,725 --> 00:15:18,830
to differentiate.
254
00:15:18,830 --> 00:15:19,940
So we're leaving it alone.
255
00:15:19,940 --> 00:15:21,689
This is the simplest
possible form for it,
256
00:15:21,689 --> 00:15:23,640
and now we're going
to differentiate.
257
00:15:23,640 --> 00:15:24,570
So what happens?
258
00:15:24,570 --> 00:15:26,640
So again what's the method?
259
00:15:26,640 --> 00:15:27,810
Let me remind you.
260
00:15:27,810 --> 00:15:30,210
You're applying d/dx
to the equation.
261
00:15:30,210 --> 00:15:33,640
So you have to differentiate
the left side of the equation,
262
00:15:33,640 --> 00:15:35,810
and differentiate the
right side of the equation.
263
00:15:35,810 --> 00:15:51,100
So it's this, and what you get
is 2x + 2yy' is equal to what?
264
00:15:51,100 --> 00:15:52,760
0.
265
00:15:52,760 --> 00:15:56,390
The derivative of 1 0.
266
00:15:56,390 --> 00:15:58,630
So this is the chain rule again.
267
00:15:58,630 --> 00:16:00,390
I did it a different way.
268
00:16:00,390 --> 00:16:02,690
I'm trying to get you used
to many different notations
269
00:16:02,690 --> 00:16:04,420
at once.
270
00:16:04,420 --> 00:16:05,350
Well really just two.
271
00:16:05,350 --> 00:16:10,600
Just the prime notation
and the dy/dx notation.
272
00:16:10,600 --> 00:16:14,370
And this is what I get.
273
00:16:14,370 --> 00:16:19,670
So now all I have to
do is solve for y'.
274
00:16:19,670 --> 00:16:24,270
So that y', if I put the 2x
on the other side, is -2x,
275
00:16:24,270 --> 00:16:27,760
and then divide by
2y, which is -x/y.
276
00:16:30,630 --> 00:16:34,600
So let's compare our
solutions, and I'll apologize,
277
00:16:34,600 --> 00:16:39,080
I'm going to have to erase
something to do that.
278
00:16:39,080 --> 00:16:44,480
So let's compare
our two solutions.
279
00:16:44,480 --> 00:16:46,460
I'm going to put this
underneath and simplify.
280
00:16:46,460 --> 00:16:48,880
So what was our
solution over here?
281
00:16:48,880 --> 00:16:51,500
It was 1/2(1-x^2)^(-1/2) (-2x).
282
00:16:56,720 --> 00:17:02,170
That was what we got over here.
283
00:17:02,170 --> 00:17:06,992
And that is the same thing, if I
cancel the 2's, and I change it
284
00:17:06,992 --> 00:17:08,450
back to looking
like a square root,
285
00:17:08,450 --> 00:17:11,536
that's the same thing as -x
divided by square root of 1 -
286
00:17:11,536 --> 00:17:13,960
x^2.
287
00:17:13,960 --> 00:17:18,380
So this is the formula
for the derivative
288
00:17:18,380 --> 00:17:21,340
when I do it the explicit way.
289
00:17:21,340 --> 00:17:29,550
And I'll just compare them,
these two expressions here.
290
00:17:29,550 --> 00:17:32,630
And notice they are the same.
291
00:17:32,630 --> 00:17:37,860
They're the same, because y
is equal to square root of 1 -
292
00:17:37,860 --> 00:17:40,230
x^2.
293
00:17:40,230 --> 00:17:40,730
Yeah?
294
00:17:40,730 --> 00:17:41,230
Question?
295
00:17:41,230 --> 00:17:46,140
Student: [INAUDIBLE]
296
00:17:46,140 --> 00:17:48,530
Professor: The question is
why did the implicit method
297
00:17:48,530 --> 00:17:50,930
not give the bottom
half of the circle?
298
00:17:50,930 --> 00:17:53,200
Very good question.
299
00:17:53,200 --> 00:17:57,000
The answer to that
is that it did.
300
00:17:57,000 --> 00:17:59,520
I just didn't mention it.
301
00:17:59,520 --> 00:18:00,890
Wait, I'll explain.
302
00:18:00,890 --> 00:18:05,300
So suppose I stuck
in a minus sign here.
303
00:18:05,300 --> 00:18:08,040
I would have gotten this
with the difference, so
304
00:18:08,040 --> 00:18:10,080
with an extra minus sign.
305
00:18:10,080 --> 00:18:12,480
But then when I compared
it to what was over there,
306
00:18:12,480 --> 00:18:15,620
I would have had to have another
different minus sign over here.
307
00:18:15,620 --> 00:18:19,310
So actually both places would
get an extra minus sign.
308
00:18:19,310 --> 00:18:20,630
And they would still coincide.
309
00:18:20,630 --> 00:18:22,760
So actually the implicit
method is a little better.
310
00:18:22,760 --> 00:18:23,940
It doesn't even
notice the difference
311
00:18:23,940 --> 00:18:24,940
between the branches.
312
00:18:24,940 --> 00:18:28,930
It does the job on both
the top and bottom half.
313
00:18:28,930 --> 00:18:31,390
Another way of saying
that is that you're
314
00:18:31,390 --> 00:18:33,200
calculating the slopes here.
315
00:18:33,200 --> 00:18:35,170
So let's look at this picture.
316
00:18:35,170 --> 00:18:36,740
Here's a slope.
317
00:18:36,740 --> 00:18:39,090
Let's just take a look
at a positive value
318
00:18:39,090 --> 00:18:42,770
of x and just check the sign
to see what's happening.
319
00:18:42,770 --> 00:18:46,982
If you take a positive value
of x over here, x is positive.
320
00:18:46,982 --> 00:18:48,190
This denominator is positive.
321
00:18:48,190 --> 00:18:49,106
The slope is negative.
322
00:18:49,106 --> 00:18:52,620
You can see that
it's tilting down.
323
00:18:52,620 --> 00:18:53,960
So it's okay.
324
00:18:53,960 --> 00:18:59,520
Now on the bottom side,
it's going to be tilting up.
325
00:18:59,520 --> 00:19:01,690
And similarly what's
happening up here
326
00:19:01,690 --> 00:19:05,275
is that both x and y are
positive, and this x and this y
327
00:19:05,275 --> 00:19:06,167
are positive.
328
00:19:06,167 --> 00:19:07,250
And the slope is negative.
329
00:19:07,250 --> 00:19:10,430
On the other hand, on the bottom
side, x is still positive,
330
00:19:10,430 --> 00:19:11,780
but y is negative.
331
00:19:11,780 --> 00:19:15,168
And it's tilting up because
the denominator is negative.
332
00:19:15,168 --> 00:19:17,084
The numerator is positive,
and this minus sign
333
00:19:17,084 --> 00:19:19,050
has a positive slope.
334
00:19:19,050 --> 00:19:23,330
So it matches perfectly
in every category.
335
00:19:23,330 --> 00:19:26,920
This complicated,
however, and it's easier
336
00:19:26,920 --> 00:19:30,000
just to keep track of
one branch at a time,
337
00:19:30,000 --> 00:19:32,850
even in advanced math.
338
00:19:32,850 --> 00:19:37,590
Okay, so we only do it
one branch at a time.
339
00:19:37,590 --> 00:19:43,970
Other questions?
340
00:19:43,970 --> 00:19:47,360
Okay, so now I want to
give you a slightly more
341
00:19:47,360 --> 00:19:49,210
complicated example here.
342
00:19:49,210 --> 00:19:52,690
And indeed some
of the-- so here's
343
00:19:52,690 --> 00:19:54,490
a little more
complicated example.
344
00:19:54,490 --> 00:19:56,900
It's not going to be the
most complicated example,
345
00:19:56,900 --> 00:20:17,980
but you know it'll
be a little tricky.
346
00:20:17,980 --> 00:20:22,520
So this example, I'm going
to give you a fourth order
347
00:20:22,520 --> 00:20:23,020
equation.
348
00:20:23,020 --> 00:20:31,980
So y^4 + xy^2 - 2 = 0.
349
00:20:31,980 --> 00:20:35,320
Now it just so
happens that there's
350
00:20:35,320 --> 00:20:38,700
a trick to solving
this equation,
351
00:20:38,700 --> 00:20:41,420
so actually you can do
both the explicit method
352
00:20:41,420 --> 00:20:46,210
and the non-explicit method.
353
00:20:46,210 --> 00:20:50,379
So the explicit method
would say okay well,
354
00:20:50,379 --> 00:20:51,420
I want to solve for this.
355
00:20:51,420 --> 00:20:55,760
So I'm going to use the
quadratic formula, but on y^2.
356
00:20:55,760 --> 00:20:59,230
This is quadratic in y^2,
because there's a fourth power
357
00:20:59,230 --> 00:21:02,891
and a second power, and the
first and third powers are
358
00:21:02,891 --> 00:21:03,390
missing.
359
00:21:03,390 --> 00:21:09,940
So this is y^2 is equal to -x
plus or minus the square root
360
00:21:09,940 --> 00:21:19,570
of x^2 - 4(-2) divided by 2.
361
00:21:19,570 --> 00:21:22,790
And so this x is the b.
362
00:21:22,790 --> 00:21:29,750
This -2 is the c, and a =
1 in the quadratic formula.
363
00:21:29,750 --> 00:21:37,200
And so the formula for y is plus
or minus the square root of -x
364
00:21:37,200 --> 00:21:45,070
plus or minus the square
root x^2 + 8 divided by 2.
365
00:21:45,070 --> 00:21:47,880
So now you can see this
problem of branches,
366
00:21:47,880 --> 00:21:50,540
this happens actually
in a lot of cases,
367
00:21:50,540 --> 00:21:53,066
coming up in an elaborate way.
368
00:21:53,066 --> 00:21:54,690
You have two choices
for the sign here.
369
00:21:54,690 --> 00:21:56,610
You have two choices
for the sign here.
370
00:21:56,610 --> 00:21:59,410
Conceivably as many as four
roots for this equation,
371
00:21:59,410 --> 00:22:02,031
because it's a fourth
degree equation.
372
00:22:02,031 --> 00:22:02,780
It's quite a mess.
373
00:22:02,780 --> 00:22:06,000
You should have to check
each branch separately.
374
00:22:06,000 --> 00:22:09,180
And this really is that
level of complexity,
375
00:22:09,180 --> 00:22:11,750
and in general
it's very difficult
376
00:22:11,750 --> 00:22:17,840
to figure out the formulas
for quartic equations.
377
00:22:17,840 --> 00:22:21,630
But fortunately we're
never going to use them.
378
00:22:21,630 --> 00:22:24,830
That is, we're never going
to need those formulas.
379
00:22:24,830 --> 00:22:31,850
So the implicit
method is far easier.
380
00:22:31,850 --> 00:22:35,230
The implicit method
just says okay I'll
381
00:22:35,230 --> 00:22:38,980
leave the equation
in its simplest form.
382
00:22:38,980 --> 00:22:40,520
And now differentiate.
383
00:22:40,520 --> 00:22:47,300
So when I differentiate,
I get 4y^3 y' plus -
384
00:22:47,300 --> 00:22:50,920
now here I have to
apply the product rule.
385
00:22:50,920 --> 00:22:56,090
So I differentiate the x
and the y^2 separately.
386
00:22:56,090 --> 00:22:59,720
First I differentiate with
respect to x, so I get y^2.
387
00:22:59,720 --> 00:23:03,220
Then I differentiate with
respect to the other factor,
388
00:23:03,220 --> 00:23:04,410
the y^2 factor.
389
00:23:04,410 --> 00:23:08,950
And I get x(2 y y').
390
00:23:08,950 --> 00:23:10,440
And then the 0 gives me 0.
391
00:23:10,440 --> 00:23:16,100
So minus 0 equals 0.
392
00:23:16,100 --> 00:23:21,970
So there's the implicit
differentiation step.
393
00:23:21,970 --> 00:23:26,260
And now I just want
to solve for y'.
394
00:23:26,260 --> 00:23:32,570
So I'm going to
factor out 4y^3 + 2xy.
395
00:23:32,570 --> 00:23:35,740
That's the factor on y'.
396
00:23:35,740 --> 00:23:39,780
And I'm going to put the
y^2 on the other side.
397
00:23:39,780 --> 00:23:43,400
-y^2 over here.
398
00:23:43,400 --> 00:23:55,110
And so the formula for y' is
-y^2 divided by 4y^3 + 2xy.
399
00:23:55,110 --> 00:24:01,420
So that's the formula
for the solution.
400
00:24:01,420 --> 00:24:06,947
For the slope.
401
00:24:06,947 --> 00:24:07,780
You have a question?
402
00:24:07,780 --> 00:24:16,340
Student: [INAUDIBLE]
403
00:24:16,340 --> 00:24:18,350
Professor: So the question
is for the y would
404
00:24:18,350 --> 00:24:22,140
we have to put in what solved
for in the explicit equation.
405
00:24:22,140 --> 00:24:24,120
And the answer is
absolutely yes.
406
00:24:24,120 --> 00:24:25,280
That's exactly the point.
407
00:24:25,280 --> 00:24:30,950
So this is not a complete
solution to a problem.
408
00:24:30,950 --> 00:24:32,710
We started with an
implicit equation.
409
00:24:32,710 --> 00:24:33,990
We differentiated.
410
00:24:33,990 --> 00:24:36,660
And we got in the end,
also an implicit equation.
411
00:24:36,660 --> 00:24:39,540
It doesn't tell us what
y is as a function of x.
412
00:24:39,540 --> 00:24:43,040
You have to go back
to this formula
413
00:24:43,040 --> 00:24:45,520
to get the formula for x.
414
00:24:45,520 --> 00:24:49,310
So for example, let me
give you an example here.
415
00:24:49,310 --> 00:24:54,660
So this hides a degree of
complexity of the problem.
416
00:24:54,660 --> 00:24:58,550
But it's a degree of complexity
that we must live with.
417
00:24:58,550 --> 00:25:10,460
So for example, at x = 1, you
can see that y = 1 solves.
418
00:25:10,460 --> 00:25:16,750
That happens to be--
solves y^4 + xy^2 - 2 = 0.
419
00:25:16,750 --> 00:25:18,400
That's why I picked
the 2 actually,
420
00:25:18,400 --> 00:25:21,000
so it would be 1 + 1 - 2 = 0.
421
00:25:21,000 --> 00:25:23,060
I just wanted to have a
convenient solution there
422
00:25:23,060 --> 00:25:25,630
to pull out of my
hat at this point.
423
00:25:25,630 --> 00:25:26,670
So I did that.
424
00:25:26,670 --> 00:25:30,250
And so we now know
that when x = 1, y = 1.
425
00:25:30,250 --> 00:25:41,740
So at (1, 1) along the curve,
the slope is equal to what?
426
00:25:41,740 --> 00:25:52,200
Well, I have to plug in
here, -1^2 / (4*1^3 + 2*1*1).
427
00:25:52,200 --> 00:25:54,290
That's just plugging in
that formula over there,
428
00:25:54,290 --> 00:25:59,170
which turns out to be -1/6.
429
00:25:59,170 --> 00:26:00,670
So I can get it.
430
00:26:00,670 --> 00:26:13,940
On the other hand,
at say x = 2, we're
431
00:26:13,940 --> 00:26:32,890
stuck using this formula
star here to find y.
432
00:26:32,890 --> 00:26:37,170
Now, so let me just
make two points
433
00:26:37,170 --> 00:26:40,020
about this, which are just
philosophical points for you
434
00:26:40,020 --> 00:26:42,420
right now.
435
00:26:42,420 --> 00:26:45,427
The first is, when
I promised you
436
00:26:45,427 --> 00:26:47,010
at the beginning of
this class that we
437
00:26:47,010 --> 00:26:48,840
were going to be
able to differentiate
438
00:26:48,840 --> 00:26:53,230
any function you know, I
meant it very literally.
439
00:26:53,230 --> 00:26:56,021
What I meant is if
you know the function,
440
00:26:56,021 --> 00:26:58,020
we'll be able give a
formula for the derivative.
441
00:26:58,020 --> 00:27:00,210
If you don't know how
to find a function,
442
00:27:00,210 --> 00:27:02,490
you'll have a lot of trouble
finding the derivative.
443
00:27:02,490 --> 00:27:05,370
So we didn't make any
promises that if you
444
00:27:05,370 --> 00:27:06,830
can't find the
function you will be
445
00:27:06,830 --> 00:27:09,300
able to find the
derivative by some magic.
446
00:27:09,300 --> 00:27:10,450
That will never happen.
447
00:27:10,450 --> 00:27:12,900
And however complex
the function is,
448
00:27:12,900 --> 00:27:16,130
a root of a fourth
degree polynomial
449
00:27:16,130 --> 00:27:20,300
can be pretty complicated
function of the coefficients,
450
00:27:20,300 --> 00:27:23,990
we're stuck with this degree
of complexity in the problem.
451
00:27:23,990 --> 00:27:27,220
But the big advantage
of his method, notice,
452
00:27:27,220 --> 00:27:29,340
is that although we've
had to find star,
453
00:27:29,340 --> 00:27:31,180
we had to find
this formula star,
454
00:27:31,180 --> 00:27:34,100
and there are many other ways of
doing these things numerically,
455
00:27:34,100 --> 00:27:36,250
by the way, which
we'll learn later,
456
00:27:36,250 --> 00:27:39,940
so there's a good method
for doing it numerically.
457
00:27:39,940 --> 00:27:43,190
Although we had to find star, we
never had to differentiate it.
458
00:27:43,190 --> 00:27:46,080
We had a fast way of
getting the slope.
459
00:27:46,080 --> 00:27:48,280
So we had to know
what x and y were.
460
00:27:48,280 --> 00:27:50,980
But y' we got by an
algebraic formula,
461
00:27:50,980 --> 00:27:54,450
in terms of the values here.
462
00:27:54,450 --> 00:27:57,140
So this is very fast,
forgetting the slope,
463
00:27:57,140 --> 00:28:02,382
once you know the point. yes?
464
00:28:02,382 --> 00:28:03,840
Student: What's in
the parentheses?
465
00:28:03,840 --> 00:28:06,790
Professor: Sorry, this is-- Well
let's see if I can manage this.
466
00:28:06,790 --> 00:28:16,522
Is this the parentheses
you're talking about?
467
00:28:16,522 --> 00:28:17,022
Ah, "say".
468
00:28:17,022 --> 00:28:17,310
That says "say".
469
00:28:17,310 --> 00:28:19,185
Well, so maybe I should
put commas around it.
470
00:28:19,185 --> 00:28:24,560
But it was S A Y,
comma comma, okay?
471
00:28:24,560 --> 00:28:28,900
Well here was at x = 1.
472
00:28:28,900 --> 00:28:33,100
I'm just throwing out a value.
473
00:28:33,100 --> 00:28:34,200
Any other value.
474
00:28:34,200 --> 00:28:36,360
Actually there is one
value, my favorite value.
475
00:28:36,360 --> 00:28:39,700
Well this is easy to
evaluate right? x = 0,
476
00:28:39,700 --> 00:28:42,610
I can do it there.
477
00:28:42,610 --> 00:28:45,400
That's maybe the only one.
478
00:28:45,400 --> 00:28:55,820
The others are a nuisance.
479
00:28:55,820 --> 00:29:03,750
All right, other questions?
480
00:29:03,750 --> 00:29:06,060
Now we have to do
something more here.
481
00:29:06,060 --> 00:29:10,470
So I claimed to you that
we could differentiate
482
00:29:10,470 --> 00:29:11,580
all the functions we know.
483
00:29:11,580 --> 00:29:13,220
But really we can
learn a tremendous
484
00:29:13,220 --> 00:29:17,910
about functions which are
really hard to get at.
485
00:29:17,910 --> 00:29:20,330
So this implicit
differentiation method
486
00:29:20,330 --> 00:29:30,750
has one very, very
important application
487
00:29:30,750 --> 00:29:38,010
to finding inverse functions,
or finding derivatives
488
00:29:38,010 --> 00:29:40,650
of inverse functions.
489
00:29:40,650 --> 00:29:51,790
So let's talk about that next.
490
00:29:51,790 --> 00:29:55,700
So first, maybe we'll just
illustrate by an example.
491
00:29:55,700 --> 00:29:59,310
If you have the function y
is equal to square root x,
492
00:29:59,310 --> 00:30:04,600
for x positive, then
of course this idea
493
00:30:04,600 --> 00:30:07,170
is that we should
simplify this equation
494
00:30:07,170 --> 00:30:10,350
and we should square it so
we get this somewhat simpler
495
00:30:10,350 --> 00:30:11,860
equation here.
496
00:30:11,860 --> 00:30:14,080
And then we have a
notation for this.
497
00:30:14,080 --> 00:30:21,690
If we call f(x) equal to
square root of x, and g(y) = x,
498
00:30:21,690 --> 00:30:25,340
this is the reversal of this.
499
00:30:25,340 --> 00:30:33,150
Then the formula for g(y)
is that it should be y^2.
500
00:30:33,150 --> 00:30:48,310
And in general, if we start
with any old y = f(x),
501
00:30:48,310 --> 00:30:52,700
and we just write down, this
is the defining relationship
502
00:30:52,700 --> 00:30:57,050
for a function g, the property
that we're saying is that
503
00:30:57,050 --> 00:31:01,040
g(f(x)) has got to
bring us back to x.
504
00:31:01,040 --> 00:31:04,620
And we write that in a
couple of different ways.
505
00:31:04,620 --> 00:31:08,260
We call g the inverse of f.
506
00:31:08,260 --> 00:31:13,400
And also we call f
the inverse of g,
507
00:31:13,400 --> 00:31:15,960
although I'm going to be
silent about which variable
508
00:31:15,960 --> 00:31:19,300
I want to use, because people
mix them up a little bit,
509
00:31:19,300 --> 00:31:31,540
as we'll be doing when we
draw some pictures of this.
510
00:31:31,540 --> 00:31:32,620
So let's see.
511
00:31:32,620 --> 00:31:42,310
Let's draw pictures of
both f and f inverse
512
00:31:42,310 --> 00:31:50,260
on the same graph.
513
00:31:50,260 --> 00:32:02,130
So first of all, I'm going
to draw the graph of f(x)
514
00:32:02,130 --> 00:32:06,470
= square root of x.
515
00:32:06,470 --> 00:32:11,260
That's some shape like this.
516
00:32:11,260 --> 00:32:16,390
And now, in order to
understand what g(y) is,
517
00:32:16,390 --> 00:32:20,030
so let's do the
analysis in general,
518
00:32:20,030 --> 00:32:23,420
but then we'll draw it
in this particular case.
519
00:32:23,420 --> 00:32:31,780
If you have g(y)
= x, that's really
520
00:32:31,780 --> 00:32:34,460
just the same equation right?
521
00:32:34,460 --> 00:32:37,440
This is the equation
g(y) = x, that's y^2 = x.
522
00:32:37,440 --> 00:32:40,650
This is y = square root of x,
those are the same equations,
523
00:32:40,650 --> 00:32:43,330
it's the same curve.
524
00:32:43,330 --> 00:32:49,807
But suppose now that we wanted
to write down what g(x) is.
525
00:32:49,807 --> 00:32:51,890
In other words, we wanted
to switch the variables,
526
00:32:51,890 --> 00:32:55,650
so draw them as I said on the
same graph with the same x,
527
00:32:55,650 --> 00:32:59,800
and the same y axes.
528
00:32:59,800 --> 00:33:04,340
Then that would be, in effect,
trading the roles of x and y.
529
00:33:04,340 --> 00:33:07,670
We have to rename every
point on the graph which
530
00:33:07,670 --> 00:33:12,290
is the ordered pair (x, y), and
trade it for the opposite one.
531
00:33:12,290 --> 00:33:15,250
And when you
exchange x and y, so
532
00:33:15,250 --> 00:33:23,800
to do this, exchange
x and y, and when
533
00:33:23,800 --> 00:33:27,030
you do that, graphically
what that looks
534
00:33:27,030 --> 00:33:30,440
like is the following:
suppose you have a place here,
535
00:33:30,440 --> 00:33:33,580
and this is the x
and this is the y,
536
00:33:33,580 --> 00:33:35,030
then you want to trade them.
537
00:33:35,030 --> 00:33:39,620
So you want the y here right?
538
00:33:39,620 --> 00:33:41,360
And the x up there.
539
00:33:41,360 --> 00:33:44,400
It's sort of the opposite
place over there.
540
00:33:44,400 --> 00:33:51,130
And that is the place which is
directly opposite this point
541
00:33:51,130 --> 00:33:55,790
across the diagonal line x = y.
542
00:33:55,790 --> 00:33:58,410
So you reflect across this
or you flip across that.
543
00:33:58,410 --> 00:34:01,120
You get this other shape
that looks like that.
544
00:34:01,120 --> 00:34:10,780
Maybe I'll draw it with a
colored piece of chalk here.
545
00:34:10,780 --> 00:34:24,090
So this guy here
is y = f^(-1)(x).
546
00:34:24,090 --> 00:34:26,224
And indeed, if you
look at these graphs,
547
00:34:26,224 --> 00:34:27,390
this one is the square root.
548
00:34:27,390 --> 00:34:34,787
This one happens to be y = x^2.
549
00:34:34,787 --> 00:34:36,370
If you take this
one, and you turn it,
550
00:34:36,370 --> 00:34:39,890
you reverse the roles of
the x axis and the y axis,
551
00:34:39,890 --> 00:34:43,620
and tilt it on its side.
552
00:34:43,620 --> 00:34:51,150
So that's the picture of what an
inverse function is, and now I
553
00:34:51,150 --> 00:34:56,070
want to show you that the method
of implicit differentiation
554
00:34:56,070 --> 00:34:59,890
allows us to compute
the derivatives
555
00:34:59,890 --> 00:35:03,250
of inverse functions.
556
00:35:03,250 --> 00:35:05,100
So let me just
say it in general,
557
00:35:05,100 --> 00:35:07,310
and then I'll carry
it out in particular.
558
00:35:07,310 --> 00:35:16,390
So implicit
differentiation allows
559
00:35:16,390 --> 00:35:32,880
us to find the derivative
of any inverse function,
560
00:35:32,880 --> 00:35:53,380
provided we know the
derivative of the function.
561
00:35:53,380 --> 00:35:58,360
So let's do that for
what is an example, which
562
00:35:58,360 --> 00:36:02,510
is truly complicated and
a little subtle here.
563
00:36:02,510 --> 00:36:04,770
It has a very pretty answer.
564
00:36:04,770 --> 00:36:09,700
So we'll carry out
an example here,
565
00:36:09,700 --> 00:36:19,660
which is the function y is
equal to the inverse tangent.
566
00:36:19,660 --> 00:36:25,900
So again, for the
inverse tangent
567
00:36:25,900 --> 00:36:30,020
all of the things
that we're going to do
568
00:36:30,020 --> 00:36:32,360
are going to be
based on simplifying
569
00:36:32,360 --> 00:36:36,250
this equation by taking
the tangent of both sides.
570
00:36:36,250 --> 00:36:38,260
So, us let me remind
you by the way,
571
00:36:38,260 --> 00:36:41,780
the inverse tangent is what's
also known as arctangent.
572
00:36:41,780 --> 00:36:45,210
That's just another
notation for the same thing.
573
00:36:45,210 --> 00:36:49,770
And what we're going
to use to describe
574
00:36:49,770 --> 00:36:55,565
this function is the
equation tan y = x.
575
00:36:55,565 --> 00:36:56,940
That's what happens
when you take
576
00:36:56,940 --> 00:36:59,110
the tangent of this function.
577
00:36:59,110 --> 00:37:01,690
This is how we're
going to figure out
578
00:37:01,690 --> 00:37:19,650
what the function looks like.
579
00:37:19,650 --> 00:37:23,370
So first of all,
I want to draw it,
580
00:37:23,370 --> 00:37:26,610
and then we'll do
the computation.
581
00:37:26,610 --> 00:37:32,120
So let's make the diagram first.
582
00:37:32,120 --> 00:37:33,630
So I want to do
something which is
583
00:37:33,630 --> 00:37:35,879
analogous to what I did over
here with the square root
584
00:37:35,879 --> 00:37:38,060
function.
585
00:37:38,060 --> 00:37:43,740
So first of all, I remind
you that the tangent function
586
00:37:43,740 --> 00:37:52,850
is defined between two values
here, which are pi/2 and -pi/2.
587
00:37:52,850 --> 00:37:55,010
And it starts out
at minus infinity
588
00:37:55,010 --> 00:37:58,560
and curves up like this.
589
00:37:58,560 --> 00:38:08,050
So that's the function tan x.
590
00:38:08,050 --> 00:38:11,350
And so the one that
we have to sketch
591
00:38:11,350 --> 00:38:14,870
is this one which we
get by reflecting this
592
00:38:14,870 --> 00:38:21,780
across the axis.
593
00:38:21,780 --> 00:38:25,150
Well not the axis, the diagonal.
594
00:38:25,150 --> 00:38:33,080
This slope by the way, should
be less - a little lower here so
595
00:38:33,080 --> 00:38:37,580
that we can have it
going down and up.
596
00:38:37,580 --> 00:38:42,000
So let me show you
what it looks like.
597
00:38:42,000 --> 00:38:44,830
On the front, it's going to
look a lot like this one.
598
00:38:44,830 --> 00:38:50,620
So this one had curved
down, and so the reflection
599
00:38:50,620 --> 00:38:52,480
across the diagonal curved up.
600
00:38:52,480 --> 00:38:54,320
Here this is curving
up, so the reflection
601
00:38:54,320 --> 00:38:56,070
is going to curve down.
602
00:38:56,070 --> 00:38:58,760
It's going to look like this.
603
00:38:58,760 --> 00:39:02,360
Maybe I should, sorry,
let's use a different color,
604
00:39:02,360 --> 00:39:04,450
because it's
reversed from before.
605
00:39:04,450 --> 00:39:10,420
I'll just call it green.
606
00:39:10,420 --> 00:39:15,630
Now, the original curve
in the first quadrant
607
00:39:15,630 --> 00:39:17,990
eventually had an asymptote
which was straight up.
608
00:39:17,990 --> 00:39:24,090
So this one is going to have an
asymptote which is horizontal.
609
00:39:24,090 --> 00:39:27,260
And that level is what?
610
00:39:27,260 --> 00:39:29,950
What's the highest?
611
00:39:29,950 --> 00:39:30,780
It is just pi/2.
612
00:39:33,470 --> 00:39:35,910
Now similarly, the
other way, we're
613
00:39:35,910 --> 00:39:40,040
going to do this:
and this bottom level
614
00:39:40,040 --> 00:39:42,940
is going to be -pi/2.
615
00:39:42,940 --> 00:39:47,480
So there's the picture
of this function.
616
00:39:47,480 --> 00:39:50,300
It's defined for all x.
617
00:39:50,300 --> 00:39:57,530
So this green guy
is y = arctan x.
618
00:39:57,530 --> 00:39:59,530
And it's defined all the
way from minus infinity
619
00:39:59,530 --> 00:40:05,290
to infinity.
620
00:40:05,290 --> 00:40:11,410
And to use a notation that
we had from limit notation
621
00:40:11,410 --> 00:40:21,290
as x goes to infinity, let's
say, x is equal to pi/2.
622
00:40:21,290 --> 00:40:24,823
That's an example of one value
that's of interest in addition
623
00:40:24,823 --> 00:40:28,250
to the finite values.
624
00:40:28,250 --> 00:40:31,430
Okay, so now the
first ingredient
625
00:40:31,430 --> 00:40:34,580
that we're going
to need, is we're
626
00:40:34,580 --> 00:40:37,190
going to need the derivative
of the tangent function.
627
00:40:37,190 --> 00:40:40,060
So I'm going to recall
for you, and maybe you
628
00:40:40,060 --> 00:40:43,180
haven't worked this out yet, but
I hope that many of you have,
629
00:40:43,180 --> 00:40:48,790
that if you take the derivative
with respect to y of tan y.
630
00:40:48,790 --> 00:40:55,500
So this you do by
the quotient rule.
631
00:40:55,500 --> 00:40:59,150
So this is of the
form u/v, right?
632
00:40:59,150 --> 00:41:00,630
You use the quotient rule.
633
00:41:00,630 --> 00:41:06,090
So I'm going to get this.
634
00:41:06,090 --> 00:41:09,560
But what you get in the end is
some marvelous simplification
635
00:41:09,560 --> 00:41:12,580
that comes out to cos^2 y.
636
00:41:12,580 --> 00:41:14,740
1 over cosine squared.
637
00:41:14,740 --> 00:41:17,460
You can recognize the cosine
squared from the fact that you
638
00:41:17,460 --> 00:41:19,240
should get v^2 in
the denominator,
639
00:41:19,240 --> 00:41:26,630
and somehow the numerators all
cancel and simplifies to 1.
640
00:41:26,630 --> 00:41:32,560
This is also known
as secant squared y.
641
00:41:32,560 --> 00:41:38,010
So that something that
if you haven't done yet,
642
00:41:38,010 --> 00:41:48,450
you're going to have to
do this as an exercise.
643
00:41:48,450 --> 00:41:50,220
So we need that
ingredient, and now we're
644
00:41:50,220 --> 00:41:59,180
just going to
differentiate our equation.
645
00:41:59,180 --> 00:42:00,540
And what do we get?
646
00:42:00,540 --> 00:42:15,460
We get, again, (d/dy tan y)
times dy/dx is equal to 1.
647
00:42:15,460 --> 00:42:22,930
Or, if you like, 1 / cos^2 y
times, in the other notation,
648
00:42:22,930 --> 00:42:30,090
y', is equal to 1.
649
00:42:30,090 --> 00:42:35,740
So I've just used the formulas
that I just wrote down there.
650
00:42:35,740 --> 00:42:37,860
Now all I have to
do is solve for y'.
651
00:42:37,860 --> 00:42:44,060
It's cos^2 y.
652
00:42:44,060 --> 00:42:47,040
Unfortunately, this
is not the form
653
00:42:47,040 --> 00:42:49,610
that we ever want to
leave these things in.
654
00:42:49,610 --> 00:42:52,180
This is the same problem we
had with that ugly square root
655
00:42:52,180 --> 00:42:54,500
expression, or with
any of the others.
656
00:42:54,500 --> 00:42:58,070
We want to rewrite
in terms of x.
657
00:42:58,070 --> 00:43:05,280
Our original question was
what is d/dx of arctan x.
658
00:43:05,280 --> 00:43:08,260
Now so far we have the following
answer to that question:
659
00:43:08,260 --> 00:43:15,210
it's cos^2 (arctan x).
660
00:43:15,210 --> 00:43:31,780
Now this is a correct answer,
but way too complicated.
661
00:43:31,780 --> 00:43:33,180
Now that doesn't
mean that if you
662
00:43:33,180 --> 00:43:35,100
took a random
collection of functions,
663
00:43:35,100 --> 00:43:37,560
you wouldn't end up with
something this complicated.
664
00:43:37,560 --> 00:43:41,000
But these particular functions,
these beautiful circular
665
00:43:41,000 --> 00:43:42,900
functions involved
with trigonometry all
666
00:43:42,900 --> 00:43:45,610
have very nice formulas
associated with them.
667
00:43:45,610 --> 00:43:48,740
And this simplifies
tremendously.
668
00:43:48,740 --> 00:43:50,360
So one of the
skills that you need
669
00:43:50,360 --> 00:43:54,660
to develop when you're
dealing with trig functions
670
00:43:54,660 --> 00:43:56,690
is to simplify this.
671
00:43:56,690 --> 00:44:03,620
And so let's see now that
expressions like this all
672
00:44:03,620 --> 00:44:07,950
simplify.
673
00:44:07,950 --> 00:44:10,400
So here we go.
674
00:44:10,400 --> 00:44:12,264
There's only one
formula, one ingredient
675
00:44:12,264 --> 00:44:14,180
that we need to use to
do this, and then we're
676
00:44:14,180 --> 00:44:15,460
going to draw a diagram.
677
00:44:15,460 --> 00:44:18,530
So the ingredient again, is the
original defining relationship
678
00:44:18,530 --> 00:44:22,150
that tan y = x.
679
00:44:22,150 --> 00:44:27,780
So tan y = x can be
encoded in a right triangle
680
00:44:27,780 --> 00:44:34,520
in the following way: here's
the right triangle and tan
681
00:44:34,520 --> 00:44:38,590
y means that y should be
represented as an angle.
682
00:44:38,590 --> 00:44:40,820
And then, its
tangent is the ratio
683
00:44:40,820 --> 00:44:43,500
of this vertical to
this horizontal side.
684
00:44:43,500 --> 00:44:46,250
So I'm just going to pick
two values that work,
685
00:44:46,250 --> 00:44:48,990
namely x and 1.
686
00:44:48,990 --> 00:44:51,930
Those are the simplest ones.
687
00:44:51,930 --> 00:44:57,670
So I've encoded this
equation in this picture.
688
00:44:57,670 --> 00:45:01,560
And now all I have to do is
figure out what the cosine of y
689
00:45:01,560 --> 00:45:03,540
is in this right triangle here.
690
00:45:03,540 --> 00:45:06,081
In order to do that, I need to
figure out what the hypotenuse
691
00:45:06,081 --> 00:45:13,280
is, but that's just
square root of 1 + x^2.
692
00:45:13,280 --> 00:45:18,580
And now I can read off
what the cosine of y is.
693
00:45:18,580 --> 00:45:23,560
So the cosine of y is 1
divided by the hypotenuse.
694
00:45:23,560 --> 00:45:32,480
So it's 1 over square root,
whoops, yeah, 1 + x^2.
695
00:45:32,480 --> 00:45:39,900
And so cosine squared
is just 1 / 1 + x^2.
696
00:45:39,900 --> 00:45:43,050
And so our answer over here, the
preferred answer which is way
697
00:45:43,050 --> 00:45:45,670
simpler than what
I wrote up there,
698
00:45:45,670 --> 00:46:04,690
is that d/dx of tan inverse
x is equal to 1 over 1 + x^2.
699
00:46:04,690 --> 00:46:06,770
Maybe I'll stop here
for one more question.
700
00:46:06,770 --> 00:46:10,240
I have one more calculation
which I can do even
701
00:46:10,240 --> 00:46:11,500
in less than a minute.
702
00:46:11,500 --> 00:46:16,360
So we have a whole
minute for questions.
703
00:46:16,360 --> 00:46:20,480
Yeah?
704
00:46:20,480 --> 00:46:26,620
Student: [INAUDIBLE]
705
00:46:26,620 --> 00:46:34,210
Professor: What happens
to the inverse tangent?
706
00:46:34,210 --> 00:46:41,530
The inverse tangent--
Okay, this inverse tangent
707
00:46:41,530 --> 00:46:44,030
is the same as this y here.
708
00:46:44,030 --> 00:46:46,140
Those are the same thing.
709
00:46:46,140 --> 00:46:50,480
So what I did was I skipped
this step here entirely.
710
00:46:50,480 --> 00:46:52,070
I never wrote that down.
711
00:46:52,070 --> 00:46:54,560
But the inverse
tangent was that y.
712
00:46:54,560 --> 00:46:56,690
The issue was what's
a good formula
713
00:46:56,690 --> 00:47:01,080
for cos y in terms of x?
714
00:47:01,080 --> 00:47:04,510
So I am evaluating that, but
I'm doing it using the letter y.
715
00:47:04,510 --> 00:47:06,600
So in other words, what
happened to the inverse
716
00:47:06,600 --> 00:47:10,350
tangent is that I
called it y, which
717
00:47:10,350 --> 00:47:15,340
is what it's been all along.
718
00:47:15,340 --> 00:47:17,450
Okay, so now I'm
going to do the case
719
00:47:17,450 --> 00:47:20,400
of the sine, the inverse sine.
720
00:47:20,400 --> 00:47:22,890
And I'll show you
how easy this is
721
00:47:22,890 --> 00:47:27,420
if I don't fuss with-- because
this one has an easy trig
722
00:47:27,420 --> 00:47:29,760
identity associated with it.
723
00:47:29,760 --> 00:47:37,870
So if y = sin^(-1)
x, and sin y = x,
724
00:47:37,870 --> 00:47:40,970
and now watch how simple it is
when I do the differentiation.
725
00:47:40,970 --> 00:47:42,440
I just differentiate.
726
00:47:42,440 --> 00:47:50,780
I get (cos y) y' = 1.
727
00:47:50,780 --> 00:48:00,580
And then, y', so that
implies that = 1 / cos y,
728
00:48:00,580 --> 00:48:03,080
and now to rewrite
that in terms of x,
729
00:48:03,080 --> 00:48:10,550
I have to just recognize that
this is the same as this,
730
00:48:10,550 --> 00:48:14,910
which is the same as 1 /
square root of 1 - x^2.
731
00:48:14,910 --> 00:48:19,600
So all told, the derivative
with respect to x of the arcsine
732
00:48:19,600 --> 00:48:30,420
function is 1 / square
root of 1 - x^2.
733
00:48:30,420 --> 00:48:32,540
So these implicit
differentiations
734
00:48:32,540 --> 00:48:34,700
are very convenient.
735
00:48:34,700 --> 00:48:38,190
However, I warn you
that you do have
736
00:48:38,190 --> 00:48:42,670
to be careful about the range of
applicability of these things.
737
00:48:42,670 --> 00:48:44,550
You have to draw a
picture like this one
738
00:48:44,550 --> 00:48:47,210
to make sure you know
where this makes sense.
739
00:48:47,210 --> 00:48:50,380
In other words, you have to pick
a branch for the sine function
740
00:48:50,380 --> 00:48:52,050
to work that out,
and there's something
741
00:48:52,050 --> 00:48:53,330
like that on your problem set.
742
00:48:53,330 --> 00:48:56,030
And it's also
discussed in your text.
743
00:48:56,030 --> 00:48:58,070
So we'll stop here.