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