Compute the second partial derivatives
∂2f/∂x2, ∂2f/∂x ∂y, ∂2f/∂y ∂x, ∂2f/∂y2
for the following function.
f(x, y) = log(x-y),
on the region where
(x, y) ≠ (0, 0)
verify the following theorem in this case.
if f(x, y) is of class c2 (is twice continuously differentiable), then the mixed partial derivatives are equal; that is,
∂^2f/∂x ∂y = ∂2f/∂y ∂x.

a) ∂2f/∂x2,

b) ∂2f/∂x ∂y,

c) ∂2f/∂y ∂x,

d) ∂2f/∂y2

We want to get the second partial derivatives of the function f(x, y) = log(x - y).

Remember that:

if g(x) = log(x)

Now we can use this rule to derive our function, we will get the first partial derivatives.

Now we need to derive these again to get:

So we can see that the mixed partial derivatives are equal, meaning that the theorem is true in this case.

If you want to learn more, you can read:

link

has first-order partial derivatives

Then the second-order partial derivatives are