  This paper proves that the approximation of pointwise derivatives of order
$s$ of functions in Sobolev space $W_2^m(\R^d)$ by linear combinations of
function values cannot have a convergence rate better than $m-s-d/2$, no matter
how many nodes are used for approximation and where they are placed. These
convergence rates are attained by {\em scalable} approximations that are exact
on polynomials of order at least $\lfloor m-d/2\rfloor +1$, proving that the
rates are optimal for given $m,\,s,$ and $d$. And, for a fixed node set
$X\subset\R^d$, the convergence rate in any Sobolev space $W_2^m(\Omega)$
cannot be better than $q-s$ where $q$ is the maximal possible order of
polynomial exactness of approximations based on $X$, no matter how large $m$
is. In particular,scalable stencil constructions via polyharmonic kernels are
shown to realize the optimal convergence rates, and good approximations of
their error in Sobolev space can be calculated via their error in Beppo-Levi
spaces. This allows to construct near-optimal stencils in Sobolev spaces stably
and efficiently, for use in meshless methods to solve partial differential
equations via generalized finite differences (RBF-FD). Numerical examples are
included for illustration.
