Faà di Bruno's Formula/Proof 3

Proof
We have that:
 * $\dfrac {D_x^k u} {k!}$ is the coefficient of $z^k$ in $u \left({x + z}\right)$
 * $\dfrac {D_u^j w} {j!}$ is the coefficient of $y^j$ in $w \left({u + y}\right)$.

Hence the coefficient of $z^n$ in $w \left({u \left({x + z}\right)}\right)$ is:


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