  In applied mathematics, especially in optimization, functions are often only
provided as so called "Black-Boxes" provided by software packages, or very
complex algorithms, which make automatic differentation very complicated or
even impossible. Hence one seeks the numerical approximation of the derivative.
  Unfortunately numerical differentation is a difficult task in itself, and it
is well known that it is numerical instable. There are many works on this
topic, including the usage of (global) Chebyshev approximations. Chebyshev
approximations have the great property that they converge very fast, if the
function is smooth. Nevertheless those approches have several drawbacks, since
in practice functions are not smooth, and a global approximation needs many
function evalutions.
  Nevertheless there is hope. Since functions in real world applications are
most times smooth except for finite points, corners or edges. This motivates to
use a local Chebyshev approach, where the function is only approximated
locally, and hence the Chebyshev approximations still yields a fast
approximation of the desired function. We will study such an approch in this
work, and will provide a numerical example
