  We propose a definition of differential operators of an associative algebra
$A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as
the zero cohomology of a certain bicomplex formed by Hom-spaces
$\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure
of a planar prop, i.e. each differential operator has multiple inputs and
outputs and they can be composed along planar graphs. Furthermore, for a
formally smooth algebra we have the surjective symbol map from $D(A)$ to the
space of poly-derivations. We also consider another planar prop $E(A)$
generated by automorphisms of the trivial associative deformation of $A$ over
the completion of a free associative algebra. We construct a natural map from
$E(A)$ to $D(A)$ and identify its image.
