What is $\frac{d^2}{dxdy}(G[x,y])^2$?

Question

What is $\frac{d^2}{dxdy}(G[x,y])^2$?

I obtain:

$$ \frac{d}{dx}\left(2G[x,y]\frac{d}{dy}G[x,y] \right)=2\frac{d}{dx}G[x,y]\frac{d}{dy}G[x,y]+2G[x,y]\frac{d^2}{dxdy}G[x,y] $$

Can this be simplified further? Does the term $2G[x,y]\frac{d^2}{dxdy}G[x,y]$ as it contains a double derivative and the other doesn't vanish to zero?

edit:

Thus, can I write:

$$ \frac{d^2}{dxdy}(G[x,y])^2=2\frac{d}{dx}G[x,y]\frac{d}{dy}G[x,y] $$

?

Answer 1

You can only delete the last term if $G$ is of the form $X(x)+Y(y)$. For example, $\frac{d^2}{dxdy}(xy)=1$.