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:
 * $\ds \map {D_x} {\prod_{m \mathop = 1}^r \paren {\dfrac {\paren {D_x^m u}^{k_m} } {k_m! \paren {m!}^{k_m} } } } = \prod_{m \mathop = 1}^r \paren {\dfrac {\paren {D_x^m u}^{k_m} } {k_m! \paren {m!}^{k_m} } } \sum_{m \mathop = 1}^r k_m \dfrac {D_x^{m + 1} u} {D_x^m u}$