WEBVTT
1
00:00:04.339 --> 00:00:06.740 A:middle L:90%
here we have a composite function, a function inside
2
00:00:06.740 --> 00:00:09.279 A:middle L:90%
of function, and the inside function is a quotient
3
00:00:09.640 --> 00:00:11.650 A:middle L:90%
. So we're going to end up using the chain
4
00:00:11.650 --> 00:00:13.929 A:middle L:90%
rule to take care of the composite and the quotient
5
00:00:13.929 --> 00:00:15.849 A:middle L:90%
rule to take care of the derivative of the insect
6
00:00:16.539 --> 00:00:19.019 A:middle L:90%
. So here we go. So first of all
7
00:00:19.019 --> 00:00:21.089 A:middle L:90%
, the derivative of the outside function, which is
8
00:00:21.089 --> 00:00:23.690 A:middle L:90%
the fifth power function, would be to bring down
9
00:00:23.690 --> 00:00:27.149 A:middle L:90%
the five and then raise the inside to the fourth
10
00:00:31.739 --> 00:00:33.600 A:middle L:90%
. Now we're going to multiply that by the derivative
11
00:00:33.600 --> 00:00:35.920 A:middle L:90%
of the inside. So here's where the quotient rule
12
00:00:35.920 --> 00:00:38.909 A:middle L:90%
comes in. So we have the bottom y squared
13
00:00:38.909 --> 00:00:41.859 A:middle L:90%
, plus one times the derivative of the top for
14
00:00:41.859 --> 00:00:44.929 A:middle L:90%
why cubed minus the top. Why did the fourth
15
00:00:44.929 --> 00:00:47.609 A:middle L:90%
plus one times the derivative of the bottom two?
16
00:00:47.609 --> 00:00:52.100 A:middle L:90%
Why over the bottom squared, Why squared plus one
17
00:00:52.109 --> 00:00:55.469 A:middle L:90%
quantity squared. And now we're going to simplify.
18
00:00:58.240 --> 00:01:00.799 A:middle L:90%
Okay, so I'm gonna write this quotient here as
19
00:01:00.810 --> 00:01:03.000 A:middle L:90%
the top to the fourth over the bottom to the
20
00:01:03.000 --> 00:01:07.549 A:middle L:90%
fourth with five times. Why did the fourth plus
21
00:01:07.549 --> 00:01:11.049 A:middle L:90%
one to the fourth over? Why squared plus one
22
00:01:11.049 --> 00:01:12.040 A:middle L:90%
to the fourth? And I do that because I
23
00:01:12.040 --> 00:01:15.280 A:middle L:90%
suspect that some things were going to simplify as we
24
00:01:15.280 --> 00:01:18.650 A:middle L:90%
go. Some things you're going to cancel. For
25
00:01:18.650 --> 00:01:19.859 A:middle L:90%
example, we can already see that the denominator of
26
00:01:19.859 --> 00:01:23.689 A:middle L:90%
the second fraction has some terms that will cancel with
27
00:01:23.689 --> 00:01:26.189 A:middle L:90%
the numerator of the first fraction we're actually, they
28
00:01:26.189 --> 00:01:29.319 A:middle L:90%
will melt. They will combine with the denominator of
29
00:01:29.319 --> 00:01:33.030 A:middle L:90%
the first fraction. Excuse me. Okay, so
30
00:01:33.030 --> 00:01:34.989 A:middle L:90%
now we want to simplify our numerator of the second
31
00:01:34.989 --> 00:01:38.280 A:middle L:90%
fraction. So we're going to distribute the four y
32
00:01:38.290 --> 00:01:41.989 A:middle L:90%
cubed, multiply it by both of those terms and
33
00:01:41.989 --> 00:01:44.700 A:middle L:90%
distribute the to y multiplied by both of those terms
34
00:01:44.709 --> 00:01:48.219 A:middle L:90%
and distribute the negative. So we have four wide
35
00:01:48.219 --> 00:01:53.370 A:middle L:90%
of the fifth power plus four y cubed minus two
36
00:01:53.370 --> 00:01:57.069 A:middle L:90%
y to the fifth power minus two. Y all
37
00:01:57.069 --> 00:01:59.579 A:middle L:90%
right. Now let's see what we can do with
38
00:01:59.579 --> 00:02:01.719 A:middle L:90%
that. Notice that we have four white of the
39
00:02:01.719 --> 00:02:05.010 A:middle L:90%
fifth minus two white of the fifth so we can
40
00:02:05.010 --> 00:02:06.939 A:middle L:90%
combine the like terms and we get to y to
41
00:02:06.939 --> 00:02:08.460 A:middle L:90%
the fifth, and we're gonna have to create a
42
00:02:08.460 --> 00:02:14.330 A:middle L:90%
little bit more space to work here. All right
43
00:02:14.330 --> 00:02:15.759 A:middle L:90%
, so this is what we just saw a moment
44
00:02:15.759 --> 00:02:17.340 A:middle L:90%
ago. And what we're doing now is we're combining
45
00:02:17.340 --> 00:02:20.150 A:middle L:90%
the four white to the fifth, minus two y
46
00:02:20.150 --> 00:02:22.900 A:middle L:90%
to the fifth. So now we have five times
47
00:02:22.900 --> 00:02:25.099 A:middle L:90%
y to the fourth plus one to the fourth times
48
00:02:25.099 --> 00:02:29.860 A:middle L:90%
two white of the fifth plus four y cubed minus
49
00:02:29.860 --> 00:02:32.770 A:middle L:90%
two y over. Now let's go ahead while we're
50
00:02:32.770 --> 00:02:36.289 A:middle L:90%
rewriting this and combined these into y squared, plus
51
00:02:36.289 --> 00:02:42.110 A:middle L:90%
one to the six power Okay, finally, let's
52
00:02:42.110 --> 00:02:44.460 A:middle L:90%
notice that each of the factors for each of the
53
00:02:44.460 --> 00:02:47.919 A:middle L:90%
terms in the the final factor of the numerator can
54
00:02:47.930 --> 00:02:51.560 A:middle L:90%
eyes a multiple of two y. So let's factor
55
00:02:51.560 --> 00:02:53.919 A:middle L:90%
to why, out of each of those and we'll
56
00:02:53.990 --> 00:02:55.400 A:middle L:90%
multiply that to why that we get factored out by
57
00:02:55.400 --> 00:02:59.349 A:middle L:90%
the five. So that will give us 10 y
58
00:03:00.280 --> 00:03:01.250 A:middle L:90%
times a white of the fourth plus one to the
59
00:03:01.250 --> 00:03:06.979 A:middle L:90%
fourth times. Why did the fourth plus two y
60
00:03:06.979 --> 00:03:10.599 A:middle L:90%
squared minus one and that's still over? Why squared
61
00:03:10.599 --> 00:03:13.169 A:middle L:90%
plus one to the six power