Faà di Bruno's Formula/Proof 1

Proof
Let $c \left({n, j, k_1, k_2, \ldots}\right)$ be the coefficient of $\left({D_x^n u}\right)^{k_n}$.

By differentiating $x$:

The equations:
 * $k_1 + k_2 + \cdots = j$

and:
 * $k_1 + 2 k_2 + \cdots = n$

are preserved by this induction step.

Thus it is possible to factor out:
 * $\dfrac {n!} {k_1! \left({1!}\right)^{k_1} \cdots k_n! \left({n!}\right)^{k_n} }$

from each term on the of the equation for $c \left({n + 1, j, k_1, k_2, \ldots}\right)$.

Thus we are left with:
 * $k_1 + 2 k_2 + 3 k_3 + \cdots = n + 1$

Note that while there are only finitely many $k$'s, it is convenient to consider an infinite sum because $k_{n + 1} = k_{n + 2} = \cdots = 0$.