  There exist many applications where it is necessary to approximate
numerically derivatives of a function which is given by a computer procedure.
In particular, all the fields of optimization have a special interest in such a
kind of information. In this paper, a new way to do this is presented for a new
kind of a computer -- the Infinity Computer -- able to work numerically with
finite, infinite, and infinitesimal numbers. It is proved that the Infinity
Computer is able to calculate values of derivatives of a higher order for a
wide class of functions represented by computer procedures. It is shown that
the ability to compute derivatives of arbitrary order automatically and
accurate to working precision is an intrinsic property of the Infinity Computer
related to its way of functioning. Numerical examples illustrating the new
concepts and numerical tools are given.
