  It is proved that one cannot approximate stably the first derivative of a
smooth function given noisy values of this function and a bound on this
function and its first derivative. Such an approximation is shown to be
possible if an a priori bound is known for a fractional derivative of order
greater than one. An algorithm is proposed for such a stable approximation and
error estimates for the proposed algorithm are given.
  Under certain assumptions it is proved that this algorithm is best possible
among all linear and nonlinear algorithms.
