  It is shown phenomenologically that the fractional derivative $\xi=D^\alpha
u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto
|\xi| ^{-p_\star}$ in its cumulative probability, for a suitable range of
$\alpha$'s. The exponent is determined by the condition $\zeta_{p_\star} =
\alpha p_\star$, where $\zeta_p$ is the exponent of the structure function of
order $p$. A detailed study is made for the case of random multiplicative
processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable
to both theory and numerical simulations. Large deviations theory provides a
concrete criterion, which involves the departure from straightness of the
$\zeta_p$ graph, for the presence of power-law tails when there is only a
limited range over which the data possess scaling properties (e.g. because of
the presence of a viscous cutoff). The method is also applied to wind tunnel
data and financial data.
