WEBVTT
1
00:00:00.540 --> 00:00:03.790 A:middle L:90%
So if we want to find why, Prime,
2
00:00:03.799 --> 00:00:06.820 A:middle L:90%
um, we could try to take the derivative of
3
00:00:06.820 --> 00:00:11.160 A:middle L:90%
this directly. But one thing I want to do
4
00:00:11.169 --> 00:00:15.849 A:middle L:90%
is first apply E on each side and you'll maybe
5
00:00:15.849 --> 00:00:17.460 A:middle L:90%
see why I want to do this in a moment
6
00:00:17.940 --> 00:00:20.910 A:middle L:90%
. So this is going to give us e to
7
00:00:20.910 --> 00:00:24.660 A:middle L:90%
the y is equal to X squared plus y squared
8
00:00:25.640 --> 00:00:29.530 A:middle L:90%
. All right, so now what I'm going to
9
00:00:29.530 --> 00:00:31.359 A:middle L:90%
do is go ahead and take the derivative here.
10
00:00:31.940 --> 00:00:34.600 A:middle L:90%
So we have deep I d. X so on
11
00:00:34.600 --> 00:00:36.619 A:middle L:90%
the left side will have to apply chain rule to
12
00:00:36.619 --> 00:00:40.520 A:middle L:90%
this eso It would be e to the y times
13
00:00:40.520 --> 00:00:43.579 A:middle L:90%
the derivative of why, which is just why prime
14
00:00:43.590 --> 00:00:45.549 A:middle L:90%
And then over here, we'd end up with two
15
00:00:45.549 --> 00:00:48.520 A:middle L:90%
x plus, uh, in the derivative of y
16
00:00:48.520 --> 00:00:50.130 A:middle L:90%
squared will have to use chain rules. Well,
17
00:00:50.130 --> 00:00:52.509 A:middle L:90%
so would be to y times the derivative of why
18
00:00:52.509 --> 00:00:56.090 A:middle L:90%
? Which is why prime And now we can go
19
00:00:56.090 --> 00:00:58.179 A:middle L:90%
ahead and get our white crimes on the same side
20
00:00:58.179 --> 00:01:00.649 A:middle L:90%
. So I'm going to subtract this over so it
21
00:01:00.649 --> 00:01:04.480 A:middle L:90%
be e to the Y times Why prime minus two
22
00:01:04.480 --> 00:01:08.939 A:middle L:90%
y y prime is equal to x then we can
23
00:01:08.939 --> 00:01:11.659 A:middle L:90%
factor out the white primes. We get white prime
24
00:01:14.040 --> 00:01:15.870 A:middle L:90%
, Um, times e to the y minus two
25
00:01:15.870 --> 00:01:19.469 A:middle L:90%
y is equal to two X. Then we could
26
00:01:19.469 --> 00:01:22.709 A:middle L:90%
go ahead and divide this over. So we did
27
00:01:22.709 --> 00:01:29.269 A:middle L:90%
it with why Prime is equal to two X over
28
00:01:29.280 --> 00:01:32.989 A:middle L:90%
e to the Y, minus two y. And
29
00:01:32.989 --> 00:01:34.250 A:middle L:90%
so now this, I believe, looks a little
30
00:01:34.250 --> 00:01:38.170 A:middle L:90%
bit different from what they give us the answer in
31
00:01:38.170 --> 00:01:42.549 A:middle L:90%
the back of the book. Um, so let's
32
00:01:42.549 --> 00:01:44.359 A:middle L:90%
go ahead and rewrite this so we can get it
33
00:01:44.359 --> 00:01:48.290 A:middle L:90%
toe look more like that solution. So, um
34
00:01:48.299 --> 00:01:49.950 A:middle L:90%
, in the solution, they have this to extra
35
00:01:49.950 --> 00:01:52.769 A:middle L:90%
number in the minus two. Why there? So
36
00:01:52.769 --> 00:01:53.950 A:middle L:90%
all we really need to do is somehow get rid
37
00:01:53.950 --> 00:01:56.620 A:middle L:90%
of this e to the Y. Well, if
38
00:01:56.620 --> 00:02:00.299 A:middle L:90%
we come back up to here, notice how we
39
00:02:00.299 --> 00:02:02.409 A:middle L:90%
have eat the Y is equal to x squared plus
40
00:02:02.409 --> 00:02:05.560 A:middle L:90%
y squared. So we could come down here and
41
00:02:05.560 --> 00:02:08.000 A:middle L:90%
just plug that in. So the why Prime is
42
00:02:08.000 --> 00:02:13.710 A:middle L:90%
equal to two x over X squared plus y squared
43
00:02:13.719 --> 00:02:17.030 A:middle L:90%
minus two by which is exactly what we were given
44
00:02:17.039 --> 00:02:21.080 A:middle L:90%
. So I mean, both of these are valid
45
00:02:21.080 --> 00:02:23.319 A:middle L:90%
solutions I mean, the only reason why I really
46
00:02:23.319 --> 00:02:25.250 A:middle L:90%
rewrote it in this way is toe make it look
47
00:02:25.250 --> 00:02:28.659 A:middle L:90%
more like the answer in the back of the book
48
00:02:29.039 --> 00:02:30.770 A:middle L:90%
. Um, but you can write it is either
49
00:02:30.770 --> 00:02:31.960 A:middle L:90%
one of these and these are still valid solutions.
50
00:02:32.939 --> 00:02:35.840 A:middle L:90%
Um, if you were to have just taken the
51
00:02:35.840 --> 00:02:38.310 A:middle L:90%
derivative directly. So like, if we didn't do
52
00:02:38.310 --> 00:02:40.990 A:middle L:90%
this exponentially ation of e on each side and just
53
00:02:40.990 --> 00:02:45.789 A:middle L:90%
did the derivative directly, then you would have got
54
00:02:45.800 --> 00:02:49.090 A:middle L:90%
this second one a little bit more straightforward. Um
55
00:02:49.099 --> 00:02:51.729 A:middle L:90%
, but doing it there was a lot more algebra
56
00:02:51.729 --> 00:02:53.009 A:middle L:90%
involved. So that's why I prefer to do it
57
00:02:53.020 --> 00:02:54.289 A:middle L:90%
this way. So, yeah, I encourage you
58
00:02:54.289 --> 00:02:57.860 A:middle L:90%
try to do it the first way more directly,
59
00:02:58.139 --> 00:03:00.159 A:middle L:90%
and then you would see that. Oh, yeah
60
00:03:00.539 --> 00:03:02.360 A:middle L:90%
. Doing it that way is a lot more work
61
00:03:02.939 --> 00:03:05.289 A:middle L:90%
. Yeah, I mean, Either way, though
62
00:03:05.300 --> 00:03:06.550 A:middle L:90%
, you get the same answer