00:01
Okay, we want to find a taylor or mclaurin series for this.
00:04
And remember, mclaurin series is just a special type of a taylor series where a is zero.
00:14
Okay, so i've worked this three times already.
00:18
The only way i can think of to do it is to just use the definition.
00:22
And it's pretty messy, but let's just take some derivatives and see what happens.
00:29
Okay, so one -half times four plus x squared to the minus one -half.
00:34
Times 2x, which gives you x over 4 plus x squared to the 1 half.
00:43
Okay, now i'm going to use the quotient rule.
00:46
It's the bottom times the derivative of the top, which is 1, minus the top, times the derivative of the bottom, all over the bottom squared.
01:07
All right, so that is 4 plus x squared to the 1ā2, minus x squared over 4 plus x squared to the 1ā2.
01:20
All over 4 plus x squared to the 1.
01:27
All right, so what i'm going to do is i'm going to multiply everything by the least common denominator.
01:43
So i get 4 plus x squared minus x squared, all over 4 plus x squared to the 3 halves, which is 4 times 4 plus x squared to the minus 3 halves.
02:00
Okay, this one i went ahead, i moved it up because there's no extra x.
02:05
I can just take the derivative the way it is.
02:10
So i'm on the third, oops, third derivative.
02:15
So it's four times negative three halves, four plus x squared to the minus five halves times two x.
02:26
So minus 12x over four plus x squared to the five halves.
02:34
Okay, so i'm going to write it like a fraction and use the quotient rule on it.
02:40
Okay, why don't i just leave it up there as to the mind? minus one half and use the product rule.
02:44
I think it's a lot easier to simplify when you use the quotient rule like this, when you got those fraction exponents.
02:52
So it's the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared.
03:15
This time we don't have any negative exponents.
03:18
Notice that i can factor a 4 plus x squared to the three halves out here.
03:27
And minus 12.
03:31
So when i factor out the minus 12 and this to the three halves, that leaves me 4 plus x squared there...