Faà di Bruno's Formula/Proof 4

Proof
$D_x^n$ can be expressed as a determinant:


 * $D_x^n = \begin{vmatrix}

\dbinom {n - 1} 0 u_1 & \dbinom {n - 1} 1 u_2 & \dbinom {n - 1} 2 u_3 & \cdots & \dbinom {n - 1} {n - 2} u_{n - 1} & \dbinom {n - 1} {n - 1} u_n      \\ -1 & \dbinom {n - 2} 0 u_1 & \dbinom {n - 2} 1 u_2 & \cdots & \dbinom {n - 2} {n - 3} u_{n - 2} & \dbinom {n - 2} {n - 2} u_{n - 1} \\ 0 &                   -1 & \dbinom {n - 3} 0 u_1 & \cdots & \dbinom {n - 3} {n - 4} u_{n - 3} & \dbinom {n - 3} {n - 3} u_{n - 2} \\ \vdots &               \vdots &                \vdots & \ddots &                            \vdots &                            \vdots \\ 0 &                    0 &                     0 & \cdots &                                -1 &                   \dbinom 0 0 u_1 \end{vmatrix}$

where $u_j := \left({D_x^j u}\right) D_u$.

Both sides of this equation are differential operators which are to be applied to $w$.