00:01
In this question, where are asked to find all second order partial derivatives of the function f.
00:06
But to do that, we first need to find the first order partial derivatives.
00:11
To find the derivative of f with respect to x, we need to fix y and consider x as the main variable.
00:19
So basically, y is just a constant now.
00:22
And the derivative of f with respect to x equals to y cube times the derivative of x to the 5, minus the derivative of x to the 5, of x to the 7 and since y is a constant now it's derivative with 0 this equals to 5 x to the 4 y cube minus 7 x to the 6 now let's find the derivative of f with respect to y this means we are going to fix x now and consider y as the main variable the derivative of f with respect to y equals to x to the 5 times the derivative of y cube minus the derivative of x to the 7 is going to be 0.
01:10
Plus the derivative of y to the 8, and this equals to 3x to the 5 y squared plus 8 y to the 7.
01:25
Now let's find the second derivatives.
01:28
The second derivative of f with respect to x is just the derivative of the previous of f x with respect to x and equals to 5 times 4 x cube y -cube minus 7 times 6 x to the 5.
01:51
Equals to 20 x cubed y cube minus 42 x to the 5.
01:59
Now the second derivative of f with respect to y is 3x to the 5 times 2y plus 8 times 7 y to 6 equals to 6 x to the 5 plus 56 y to the 6...