00:01
I'm going to take a look at the function r of x, y, equals 6y over x plus y.
00:08
And i want to find all second order partial derivatives.
00:13
So with that means i need to find the first derivative with respect to x and the first derivative with respect to y.
00:20
And then go through it again.
00:24
The catch there is, as i do my first derivative with respect to x, my first derivative with respect to y, as i do the second, i could do the second with x or i could do the second with y.
00:34
So i'm actually going to see that this has four different final answers for the second derivative because i have four different combinations of the variables i could look at as my variable term here for the derivative.
00:49
So i'm going to try and find rxx, right, the second with respect x both times, and then x, y, x and then y, x and then y, x and y, so looking at my first derivative with respect to x, so this is the r sub x, that partial derivative, i see that i need to do a quotient roll.
01:15
And i would do low derivative of the top is just a constant, minus top derivative of the denominator all over that denominator squared.
01:26
So really my first derivative with respect to x is negative 6y.
01:33
Over x plus y quantity squared.
01:37
Now i can go through and find that second derivative with respect to x, and i can find the second derivative with respect to y.
01:49
So for my first one, i'm going to do the second derivative with respect to x.
01:54
Now starting from my first derivative here.
01:57
So again, i can think of this as a quotient rule or even think of it as a form of the power rule.
02:03
Here, if i were to think of it like negative 6y times x plus y to the negative second.
02:11
And maybe that power rule form is a little bit faster for me.
02:14
I bring that negative 2 in front to get 12 y.
02:18
And then i need to decrease the exponent by 1.
02:21
So that's going to make it an x plus y to the negative 3rd or just say x plus y cubed.
02:26
I could get that same answer though by going through and finding the quotient rule here as well.
02:32
So that's my second derivative in terms of x.
02:37
Starting with that same first derivative, i can also write the second derivative in terms of y.
02:47
So again, starting with negative 6y over x plus y squared.
02:51
And i definitely want to go through and do the quotient rule here.
02:55
So denominator, derivative of the numerator, minus the numerator, derivative of the denominator, derivative of the denominator.
03:07
And then that derivative of the inside would just be a 1 from the y over the denominator squared would become an x plus y to the fourth.
03:17
If i can make that right a little bit nicer there.
03:21
So then there's a little bit we can do to maybe simplify, get this looking nicer than what it is.
03:28
I noticed that actually all three terms here, i'm thinking one, two, three as i say that, all three of those terms have an x plus y value that i could reduce out...