Universality of methods approximating the derivative

We prove the existence of universal functions for mappings $T_n : C ([0,1]) \to L^p ([0,1]), 0 < p < 1$, with $T_n (f) \to f' (n \to \infty)$ on certain subsets of $C^1 ([0,1])$. As an application we conclude that there are continuous functions $f \in C ([0,1])$, such that the derivatives of the Bernstein polynomials
$$ \{ (B_n (f))' : n \in \N $$
form a dense subset of $L^p ([0,1])$ for each $0<p<1$.