Is the partial derivative of a sum always the sum of partial derivatives

Question

What I mean is, if we want to find the partial deriative

$\frac{\partial^2}{\partial x \partial y} (f(x,y)+g(x,y))$

Then is it always true that:

$\frac{\partial^2}{\partial x \partial y} (f(x,y)+g(x,y))=\frac{\partial^2}{\partial x \partial y} f(x,y) + \frac{\partial^2}{\partial x \partial y} g(x,y)$

and more over can this be extended to any finite sum?

Answer 1

Yes it is true (assuming that the RHS exists and you don't have indeterminate form). However, the LHS might exist even though RHS does not exist or it is indeterminate.