Faà di Bruno's Formula

Theorem
Let $D_x^k u$ denote the $k$th derivative of a function $u$ {{{WRT|Differentiation}} $x$.

Then:
 * $\displaystyle D_x^n w = \sum_{j \mathop = 0}^n \sum_{\substack {k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_n \mathop = j \\ k_1 \mathop + 2 k_2 \mathop + \mathop \cdots \mathop + n k_n \mathop = n \\ k_1, k_2, \ldots, k_n \mathop \ge 0} } D_u^j w \dfrac {n!} {k_1! \left({1!}\right)^{k_1} \cdots k_n! \left({n!}\right)^{k_n} } \left({D_x^1 u}\right)^{k_1} \cdots \left({D_x^n u}\right)^{k_n}$