I know that $F(x,y)= \mathbb P [X \leq x, Y \leq y] = \int^x_{-\infty}\int^y_{-\infty}f_{X,Y}(u,v)dvdu$, is differentiated with respect to $x$ and $y$ to calculate the joint density $f_{X,Y}(x,y)$. I have seen in a few places that $\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = \partial^2_{xy}\mathbb P [X > x, Y \leq y] $, so that either expression can be used to calculate the joint density. I can't seem to verify if this is true or not.
I have the following relation: $\mathbb P [X > x, Y \leq y] + \mathbb P [X \leq x, Y \leq y] = \mathbb P [Y \leq y]$ from the law of total probability, and so calculating just $\partial_x$ of this expression gives:
$\partial_x \mathbb P [X > x, Y \leq y] + \partial_x \mathbb P [X \leq x, Y \leq y] = 0$. And so $\partial^2_{xy} \mathbb P [X > x, Y \leq y] + \partial^2_{xy} \mathbb P [X \leq x, Y \leq y] = 0$, which implies $\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = -\partial^2_{xy}\mathbb P [X > x, Y \leq y] $.
This seems to say that the densities from either expression gives different results. Does my logic here make sense? Any help would be appreciated. Thanks!
What you have done is correct. The second formula is wrong.