I just found a formula (Proposition 5 in http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r1) that states the product rule for a mixed partial derivative of a product of two functions. Is there something similar for a product of $n$ functions? That is,
$\frac{\partial^n}{\partial x_1 \dots \partial x_n} \prod_{i=1}^n f_i(x_1,\dots,x_n) = $ ?