How to prove this theorem about partial derivative and how to re-write this theorem with Newton's notation? i can't understand totally without newton's notation! i accept if you just generalize this from product rule.
From wikipedia section 6.3 of http://en.wikipedia.org/wiki/Product_rule:
For partial derivatives, we have
${\partial^n \over \partial x_1\,\cdots\,\partial x_n} (uv) = \sum_S {\partial^{|S|} u \over \prod_{i\in S} \partial x_i} \cdot {\partial^{n-|S|} v \over \prod_{i\not\in S} \partial x_i}$
where the index ''S'' runs through the whole list of 2''n'' subsets of {1, ..., ''n''}. For example, when ''n'' = 3, then
$\begin{align} &{}\quad {\partial^3 \over \partial x_1\,\partial x_2\,\partial x_3} (uv) \\ \\ &{}= u \cdot{\partial^3 v \over \partial x_1\,\partial x_2\,\partial x_3} + {\partial u \over \partial x_1}\cdot{\partial^2 v \over \partial x_2\,\partial x_3} + {\partial u \over \partial x_2}\cdot{\partial^2 v \over \partial x_1\,\partial x_3} + {\partial u \over \partial x_3}\cdot{\partial^2 v \over \partial x_1\,\partial x_2} \\ \\ &{}\qquad + {\partial^2 u \over \partial x_1\,\partial x_2}\cdot{\partial v \over \partial x_3} + {\partial^2 u \over \partial x_1\,\partial x_3}\cdot{\partial v \over \partial x_2} + {\partial^2 u \over \partial x_2\,\partial x_3}\cdot{\partial v \over \partial x_1} + {\partial^3 u \over \partial x_1\,\partial x_2\,\partial x_3}\cdot v. \end{align}$