  We study the following problem and its applications: given a homogeneous
degree-$d$ polynomial $g$ as an arithmetic circuit, and a $d \times d$ matrix
$X$ whose entries are homogeneous linear polynomials, compute
$g(\partial/\partial x_1, \ldots, \partial/\partial x_n) \det X$. By
considering special cases of this problem we obtain faster parameterized
algorithms for several problems, including the matroid $k$-parity and
$k$-matroid intersection problems, faster \emph{deterministic} algorithms for
testing if a linear space of matrices contains an invertible matrix (Edmonds's
problem) and detecting $k$-internal outbranchings, and more. We also match the
runtime of the fastest known deterministic algorithm for detecting subgraphs of
bounded pathwidth, while using a new approach.
  Our approach raises questions in algebraic complexity related to Waring rank
and the exponent of matrix multiplication $\omega$. In particular, we study a
new complexity measure on the space of homogeneous polynomials, namely the
bilinear complexity of a polynomial's apolar algebra. Our algorithmic
improvements are reflective of the fact that for the degree-$n$ determinant
polynomial this quantity is at most $O(n 2^{\omega n})$, whereas all known
upper bounds on the Waring rank of this polynomial exceed $n!$.
