Universality of methods approximating the derivative

We prove the existence of universal functions for mappings T-n : C([0,1]) --> L-p ([0,1]), 0 < p < 1, with T-n(f) --> f' (n --> infinity) on certain subsets of C-1 ([0,1]). As an application we conclude that there are continuous functions f E C([0,1]), such that the derivatives of the Bernstein polynomials {(B-n(f)) :n is an element of N} form a dense subset of L-p([0,1]) for each 0 < p < 1.