00:01
In the given question, we have s of x, y is equals to x to the power of 8, y to the power of 5 plus 9 x to the power of 5 and here we have y.
00:19
So we need to find all the second partial derivatives for this given function.
00:24
So this is f of xx, x, y.
00:28
So when you have double x, it means you have to partially differentiate equation 1 two times.
00:33
So one by one we just find out each and every second partial derivative.
00:39
So firstly i'll find out the first partial derivative.
00:43
So the first partial derivative with respect to x means y is constant.
00:49
So treating y as constant, the differentiation of x power of 8 will be 8 x power of 7 and y power of 5 will be written as it is.
00:58
Then the differentiation of x power 5 will be 9 in multiplication with 5 x power 4, y will be written as it is.
01:07
So thus fx, y is equal to 8 x power of 7, y power of 5 and this is 45 x power of 4, y.
01:19
Alright, now we need to differentiate this equation 2 again with respect to x.
01:24
So you write differentiating equation 2 with respect to x and partially.
01:33
So since we are integrating partially, obviously y will be treated as constant.
01:38
So thus fx, x, x, y will be equals to 8 x 7 x power of 6 and this is y power of 5.
01:50
This is 45 in multiplication with 4 x power of 3 and that's fine.
01:57
So thus fx, x, x, y over here will be 56 x power 6, y power of 5 and this will be 45 into 4, i think it will go by 170 x power of 3.
02:13
Right, so now they are asking us what is f y, x.
02:20
So f y, x over here refers that you need to differentiate equation 2 partially with respect to y.
02:28
Okay, so now you will write differentiating equation 2 partially with respect to y.
02:39
So when you will do this, you will get f of y, x is equals to.
02:44
So this means that x will be treated as constant.
02:48
So this will be 8 x power of 7 and y power of 5 differentiation will be 5 y power of 4.
02:56
Okay, now you have 45 x power of 4 written as it is differentiation of y will be 1.
03:04
So thus f of y, x, x, y is equal to 8 5 times 40 x power 7, y power of 4 plus of 45 x power of 4.
03:16
Alright, so now let's try to find out f y, x, y that is differentiating equation 1 partially with respect to y.
03:29
Okay, so i'll just write it here first and then i will write the 1.
03:33
So you will write differentiating equation 1 partially with respect to y.
03:48
So thus you will get f y, x, y is equals to x power of 8 because x will be treated as constant and differentiation of y power 5 will be 5 y power of 4 plus 9 x power 5 and differentiation of y will be how much it will be? okay, so thus f y, x, y is equals to 5 x power of 8, y power of 4 and this is 9 x power of 5...