00:01
All right, so in the black here, we have the equation we're trying to take the derivative of.
00:07
So, first things first, with these fractions, 1 over u, 1 over u squared, you know, we could do a quotient rule thing on them, but it's much easier if we rewrite these, right? one over u, okay, is the same as just u to the negative first.
00:27
Okay, if we can write our problem in terms of, you know, things with powers, u to the something, x to the something, it leaves us much better when we're taking derivatives.
00:40
Okay, so you to the negative 1 plus u to the negative 2.
00:46
All right.
00:48
Introducing the negative, let's just flip it to the other side of the fraction, okay, to the top.
00:54
Times u plus you to the negative one.
00:59
Okay, so now we've got two things, two chunks being multiplied together, right? it's this part here, u to the negative one plus u to the negative two, multiplied with u plus u to the negative one.
01:15
So when we take the derivative of this, we're going to use product rule, which is in the green there.
01:20
Okay, so our red, thing is going to be f so u to the negative one plus u to the negative two is f all right we'll need f prime and our g is going to be u plus u to the negative one okay and we will need g prime so the derivative of that all right um so what is the derivative then of you the negative one plus you to the negative two.
01:55
Well, we'll take it turn by term and we'll use the power rule.
01:58
So bring the negative one to the front.
02:01
Negative one, you, and now we lower the power by one.
02:06
So we do negative one, minus one.
02:08
That takes us down to negative two.
02:10
Okay, don't think that that makes it go away or something.
02:15
All right, let's not write plus there because in this next part, the plus you to the negative two part, we're going to bring down the negative two.
02:23
So we'll have minus two, and then we lower the power by 1, negative 3.
02:29
U to the negative 3.
02:31
Okay, so that's our f prime.
02:33
All right.
02:34
As far as g prime goes, taking the derivative of g.
02:37
All right, the derivative of u is just one, right? we bring invisible power of 1 there.
02:45
We bring it down, and then the u becomes u to the 0, which is just 1 again, so 1 times 1.
02:53
All right, and then we add, well, we don't end up adding, do we bring the negative one down so negative one you and now lower the power to the negative two so it's you know one plus i could have said one plus and then spound that this was negative one you to negative two but i'm just going to keep it as minus there all right and really we don't need this one here either or the one up here in the f prime all right so from here we've got all the that we need.
03:32
So we're just going to use this rule.
03:34
All right, so we're going to take the derivative here.
03:38
All right, and we'll start with f prime.
03:42
Okay, so f prime is negative u to the negative 2 minus 2 u to the negative 3.
03:52
Okay, and then times g, okay, so let's put this in parentheses since there's more than one term here.
04:02
And then multiply it by g, u plus u to the negative 1, all right, plus, okay, f times g prime.
04:13
So, f that was u to the negative 1 plus u to the negative 2.
04:23
We'll multiply that by g prime, 1 minus u to the negative 2.
04:30
Okay, so a lot going on here.
04:33
A lot of distributing we're going to have to do to make this look nicer.
04:36
Okay, this is our derivative, but it's not in a good form right now.
04:40
So let's distribute.
04:41
So negative u to the negative 2 times you.
04:47
Well, the coefficient is negative 1 times 1, so we'll get a negative.
04:52
All right, you to the negative 2 times you.
04:56
Remember, this is like you to the first.
04:58
Okay, so we add the exponents together when we're multiplying with the same base...
