  In the present article, a new method for the evaluation of fractional
derivatives of arbitrary real order is proposed. Numerous but inequivalent
formulations have been given in the past. Some of them exhibit unsatisfactory
properties such as e.g. the absence of a semigroup property. It is shown that
the proposed definition is free from such drawbacks. In order to achieve such a
generalization one has to work in the context of tempered distributions
(generalized functions) where the concept works nicely. The last part of the
article shows the possibility of defining fractional derivatives of periodic
functions (Fourier series).
