WEBVTT
1
00:00:03.040 --> 00:00:06.349 A:middle L:90%
Let's find the derivative of F of X and F
2
00:00:06.349 --> 00:00:09.000 A:middle L:90%
of X is a product. The first factor is
3
00:00:09.000 --> 00:00:11.310 A:middle L:90%
two X minus three to the fourth on. The
4
00:00:11.310 --> 00:00:14.189 A:middle L:90%
second factor is X squared plus X plus one to
5
00:00:14.189 --> 00:00:16.550 A:middle L:90%
the fifth. So we're going to use the product
6
00:00:16.550 --> 00:00:19.339 A:middle L:90%
rule. So we start with the 1st 2 X
7
00:00:19.339 --> 00:00:22.539 A:middle L:90%
minus three to the fourth times, the derivative of
8
00:00:22.539 --> 00:00:25.399 A:middle L:90%
the second and the second is a composite. So
9
00:00:25.399 --> 00:00:27.250 A:middle L:90%
we're going to need to use the chain rule.
10
00:00:27.640 --> 00:00:30.269 A:middle L:90%
So the derivative of the outside the outside would be
11
00:00:30.269 --> 00:00:32.700 A:middle L:90%
the fifth power function, So we would bring down
12
00:00:32.700 --> 00:00:36.049 A:middle L:90%
the five and raise the inside to the fourth.
13
00:00:36.840 --> 00:00:39.140 A:middle L:90%
So that takes care of the derivative of the outside
14
00:00:39.500 --> 00:00:41.219 A:middle L:90%
. Now we multiply it by the derivative of the
15
00:00:41.219 --> 00:00:44.229 A:middle L:90%
inside, the derivative of X squared plus X plus
16
00:00:44.229 --> 00:00:47.359 A:middle L:90%
one would be to X plus one. So what
17
00:00:47.359 --> 00:00:50.179 A:middle L:90%
we've done so far is the first times the derivative
18
00:00:50.179 --> 00:00:52.850 A:middle L:90%
of the second. Now we're gonna do plus the
19
00:00:52.850 --> 00:00:57.750 A:middle L:90%
second X squared plus X plus one to the fifth
20
00:00:58.140 --> 00:01:00.049 A:middle L:90%
times. The derivative of the first and the first
21
00:01:00.060 --> 00:01:03.320 A:middle L:90%
is also a composite. So we're going to use
22
00:01:03.320 --> 00:01:06.239 A:middle L:90%
the chain rule to find the derivative of two X
23
00:01:06.239 --> 00:01:08.000 A:middle L:90%
minus three to the fourth. The outside function is
24
00:01:08.000 --> 00:01:10.920 A:middle L:90%
the fourth power function. So we bring down the
25
00:01:10.920 --> 00:01:12.730 A:middle L:90%
four and we raised two X minus three to the
26
00:01:12.730 --> 00:01:15.459 A:middle L:90%
third. And now we multiply by the derivative of
27
00:01:15.459 --> 00:01:19.329 A:middle L:90%
the inside. The derivative of two X minus three
28
00:01:19.329 --> 00:01:22.049 A:middle L:90%
is two. Okay, so we have our derivative
29
00:01:22.060 --> 00:01:25.409 A:middle L:90%
, and now it's a matter of simplifying. So
30
00:01:25.730 --> 00:01:27.939 A:middle L:90%
this entire first part is our first term. And
31
00:01:27.939 --> 00:01:32.810 A:middle L:90%
this entire second part is our second term. Squeeze
32
00:01:32.810 --> 00:01:34.010 A:middle L:90%
my one back in there. Let's see if we
33
00:01:34.010 --> 00:01:37.849 A:middle L:90%
have any common factors that weaken factor out of both
34
00:01:37.849 --> 00:01:41.439 A:middle L:90%
terms. It looks like both terms have two X
35
00:01:41.439 --> 00:01:44.150 A:middle L:90%
minus three. This one has it to the fourth
36
00:01:44.579 --> 00:01:46.780 A:middle L:90%
. This one has it to the third so we
37
00:01:46.780 --> 00:01:49.049 A:middle L:90%
can factor out two X minus three to the third
38
00:01:49.439 --> 00:01:53.670 A:middle L:90%
from both of them. And it looks like both
39
00:01:53.670 --> 00:01:57.209 A:middle L:90%
terms have X squared plus X plus one. This
40
00:01:57.209 --> 00:01:59.709 A:middle L:90%
one has it to the fourth. This one has
41
00:01:59.709 --> 00:02:01.540 A:middle L:90%
it to the fifth so we can factor out X
42
00:02:01.540 --> 00:02:06.609 A:middle L:90%
squared plus X plus one to the fourth from both
43
00:02:06.609 --> 00:02:09.090 A:middle L:90%
of them. So then the remaining factor, what
44
00:02:09.090 --> 00:02:12.539 A:middle L:90%
we still have in the first part. We still
45
00:02:12.539 --> 00:02:14.699 A:middle L:90%
have a two X minus three. We still have
46
00:02:14.699 --> 00:02:15.990 A:middle L:90%
a five and we to still have a two X
47
00:02:15.990 --> 00:02:23.699 A:middle L:90%
plus one. So we'll put the five first five
48
00:02:23.699 --> 00:02:27.449 A:middle L:90%
times two x minus three times two x plus one
49
00:02:28.479 --> 00:02:30.020 A:middle L:90%
plus. Now let's look at what we have left
50
00:02:30.020 --> 00:02:34.909 A:middle L:90%
in the second factor the second term. So we
51
00:02:34.909 --> 00:02:37.629 A:middle L:90%
factored out four of these and we still have one
52
00:02:37.629 --> 00:02:39.000 A:middle L:90%
left. So we still have X squared plus X
53
00:02:39.000 --> 00:02:42.669 A:middle L:90%
plus one. We still have the four. We
54
00:02:42.669 --> 00:02:44.810 A:middle L:90%
still have the two, and we factored all of
55
00:02:44.810 --> 00:02:46.159 A:middle L:90%
those. So four times, too. We have
56
00:02:46.169 --> 00:02:51.949 A:middle L:90%
eight times X squared plus X plus one. Now
57
00:02:52.340 --> 00:02:54.860 A:middle L:90%
it may not be important or necessary for you to
58
00:02:54.870 --> 00:02:58.169 A:middle L:90%
do all of this simplifying. It just depends on
59
00:02:58.169 --> 00:03:00.889 A:middle L:90%
what your instructor requires, but it's good to know
60
00:03:00.889 --> 00:03:02.759 A:middle L:90%
how, just in case you need to. Okay
61
00:03:02.759 --> 00:03:05.639 A:middle L:90%
, the last thing I'm going to do is simplify
62
00:03:05.639 --> 00:03:08.270 A:middle L:90%
this and I'm going to do that by using the
63
00:03:08.270 --> 00:03:13.000 A:middle L:90%
foil method on the binomial, distributing the five and
64
00:03:13.000 --> 00:03:15.680 A:middle L:90%
then combining like terms distributing the eight as well in
65
00:03:15.680 --> 00:03:19.509 A:middle L:90%
combining like terms. So we're going to have are
66
00:03:19.509 --> 00:03:23.680 A:middle L:90%
two x minus three cubed our X squared plus X
67
00:03:23.680 --> 00:03:28.479 A:middle L:90%
plus one to the fourth. And then okay,
68
00:03:28.479 --> 00:03:31.169 A:middle L:90%
when we foil, we get four x squared and
69
00:03:31.169 --> 00:03:35.050 A:middle L:90%
we're multiplying that by five. So 20 x squared
70
00:03:36.139 --> 00:03:40.520 A:middle L:90%
we get plus two x and minus six x So
71
00:03:40.520 --> 00:03:43.620 A:middle L:90%
that's minus four X. And we're multiplying that by
72
00:03:43.620 --> 00:03:47.900 A:middle L:90%
five. So minus 20 x and we get minus
73
00:03:47.900 --> 00:03:51.319 A:middle L:90%
three and we're multiplying that by five. So minus
74
00:03:51.319 --> 00:03:54.340 A:middle L:90%
15 we distribute the eight we get eight x squared
75
00:03:54.349 --> 00:04:02.389 A:middle L:90%
eight x and eight. Okay, the last thing
76
00:04:02.389 --> 00:04:04.349 A:middle L:90%
we need to do is combine the like terms.
77
00:04:04.740 --> 00:04:06.849 A:middle L:90%
Gonna give ourselves a little bit more room to work
78
00:04:06.849 --> 00:04:13.599 A:middle L:90%
here. And here's what we just had on the
79
00:04:13.599 --> 00:04:15.269 A:middle L:90%
last step. So we're going to combine 20 x
80
00:04:15.269 --> 00:04:18.120 A:middle L:90%
squared and eight x squared, and we have 28
81
00:04:18.129 --> 00:04:23.430 A:middle L:90%
x squared. We're going to combine negative 20 x
82
00:04:23.430 --> 00:04:25.579 A:middle L:90%
and positive. Eight x. We get negative 12
83
00:04:25.579 --> 00:04:29.360 A:middle L:90%
x and we have negative 15 and positive eight.
84
00:04:29.360 --> 00:04:31.639 A:middle L:90%
So that would be minus seven. So that times
85
00:04:31.639 --> 00:04:35.680 A:middle L:90%
the other factors gives us are derivative in a simplified
86
00:04:35.680 --> A:middle L:90%
form