Suppose I have a function $f(x,y)$ and wish to find $\frac{\partial^2 f}{\partial x \partial y}$.
However, it is easier for me to do so if I use the substitutions $u = g(x)$ and $v=h(y)$
My question is: What is the rule for using substitution to find mixed partial derivatives. I wasn't able to find any looking on Google. Is it okay to just use:
$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial u \partial v} \frac{\partial u}{\partial x} \frac{\partial v}{\partial y}$ ?
The derivatives I am trying to find are very messy (asymmetric copulas for various distributions), and involve using substitutions like $u = -logx$ etc.