  We establish a new formula for the fractional derivative with Mittag-Leffler
kernel, in the form of a series of Riemann-Liouville fractional integrals,
which brings out more clearly the non-locality of fractional derivatives and is
easier to handle for certain computational purposes. We also prove existence
and uniqueness results for certain families of linear and nonlinear fractional
ODEs defined using this fractional derivative. We consider the possibility of a
semigroup property for these derivatives, and establish extensions of the
product rule and chain rule, with an application to fractional mechanics.
