Faà di Bruno's Formula/Lemma 2

Theorem
Let $m \in \Z_{\ge 1}$ be a (strictly) positive integer.

Let $k_m \in \Z_{\ge 1}$ also be a (strictly) positive integer.

Let $u: \R \to \R$ be a function of $x$ which is appropriately differentiable.

Then:
 * $\displaystyle D_x \left({\prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) }\right) = \prod_{m \mathop = 1}^r \left({\dfrac {\left({D_x^m u}\right)^{k_m} } {k_m! \left({m!}\right)^{k_m} } }\right) \sum_{m \mathop = 1}^r k_m \dfrac {D_x^{m + 1} u} {D_x^m u}$