  In this paper, we introduce some fundamental notions related to the so-called
stochastic derivatives with respect to a given $\sigma$-field $\mathcal{Q}$. In
our framework, we recall well-known results about Markov--Wiener diffusions. We
then focus mainly on the case where $X$ is a fractional diffusion and where
$\mathcal{Q}$ is the past, the future or the present of $X$. We treat some
crucial examples and our main result is the existence of stochastic derivatives
with respect to the present of $X$ when $X$ solves a stochastic differential
equation driven by a fractional Brownian motion with Hurst index $H>1/2$. We
give explicit formulas.
