I stumbled upon the following formula in a systems control textbook :
$$ s\left(\overline{x}^{(n-1)},t\right)=\left(\frac{d}{dt}+\lambda\right)^{(n-1)} e(t) \in R$$
where $\overline{x}^{(n-1)}=[x\ \dot{x} \dots x^{(n-1)} ]^T$
The textbook states that the derivative of this function wrt time is : $$ \dot{s}=\sum_{k=0}^{n-1} \binom{n-1}{k} \lambda^k e^{(n-k)} \in R$$
Can anyone understand how this is done?
This is a complete bastardization of notation, but this is just the binomial theorem. In general:
$$(a+b)^m = \sum_{i=0}^m \begin{pmatrix} m \\ i \end{pmatrix} a^ib^{m-i}.$$
Plug in the differential operator for $a$, and $\lambda$ for $b$, and set your limits accordingly in the definition of $s$. Then, the derivative follows.