I am supposed to check the below equation:
$$ \frac{\partial^3 f}{\partial x^2 \partial y} = \frac{\partial^3 f}{\partial x \partial y \partial x} = \frac{\partial^3 f}{\partial y \partial x^2} $$
where
$$ f(x,y) = x\sin^2y $$
I understand I could just calculate all of these partial derivatives and compare them, but I believe that the smarter way to do this is to use Schwarz's theorem. But I don't exactly know how. I assume I'd have to prove that $x\sin^2y$ has third order partial derivatives continuous on R. But then I would still have to manually calculate them, which defies the point of using the theorem to simply. Can someone suggest something? Thanks.