A Vandermonde-type matrix This is most certainly known, but I was unable to find a reference.
For a polynomial $P$ over a field $\mathbb F$ and a set $\{a_1,\dotsc,a_n\}$ of field elements, consider the matrix
$$ M = \begin{pmatrix}
     P(a_1) & P(a_2) & \dotsb & P(a_n) \\
     P'(a_1) & P'(a_2) & \dotsb & P'(a_n) \\
       & & \vdots \\
     P^{(n-1)}(a_1) & P^{(n-1)}(a_2) & \dotsb & P^{(n-1)}(a_n)
   \end{pmatrix} $$
where $P^{(k)}$ are the formal derivatives of $P$. When is this matrix non-singular?
 A: This is not an answer, just a long comment.
If there is no easy answer then I suggest that you need to generalise a bit before you look for an answer in the literature. My first suggestion is that (for a moment) you make replace the $a_i$ by a set of independent transcendentals $x_1,x_2, \dots$. Let $y_1,y_2,\dots$ be another such, and then let $P(X):=\prod_1^{m}(X-y_i)$ so that $P(X)=\sum_{0}^{m}f_i (-1)^{n-i}X^i$, say. 
Consider now the function
$$
\frac{\left|\begin{matrix}
     P(x_1) & P(x_2) & \dotsb & P(x_n) \\
     P'(x_1) & P'(x_2) & \dotsb & P'(x_n) \\
       & & \vdots \\
     P^{(n-1)}(x_1) & P^{(n-1)}(x_2) & \dotsb & P^{(n-1)}(x_n)
   \end{matrix}\right|}{\Delta(x_1,x_2,\dots, x_n)}
$$
where $\Delta(x_1,x_2,\dots, x_n)$ is the usual Vandermonde determinant. This is a symmetric function, and it seems a natural question to ask whether we can express it explicitly in terms of $P$ and some natural basis of the symmetric functions: perhaps there is an answer in one of the standard books, but I don't know enough to tell. 
If one were very bold one could ask whether this function is recognisable:
$$
\frac{\left|\begin{matrix}
     P^{(r_1)}(x_1) & P^{(r_1)}(x_2) & \dotsb & P^{(r_1)}(x_n) \\
     P^{(r_2)}(x_1) & P^{(r_2)}(x_2) & \dotsb & P^{(r_2)}(x_n) \\
       & & \vdots \\
     P^{(r_n)}(x_1) & P^{(r_n)}(x_2) & \dotsb & P^{(r_n)}(x_n)
   \end{matrix}\right|}{\Delta(x_1,x_2,\dots, x_n)}.
$$
