  In the first part of this paper we establish, in terms of so called
k-tangential sets, a kind of optimal estimate for the size and structure of the
set of non-differentiability of Lipshitz functions with one-sided directional
derivatives. These results can be applied to many important special functions
in analysis, like distance functions or different maximal functions. In the
second part, having the results from the first part in our use, we focus more
carefully on the differentiability properties of the classical Hardy-Littlewood
maximal function. For example, we will show that if f is continuous and
differentiable outside a countable union of tangential sets, then the same
holds to the maximal function Mf as well (if Mf is not identically infinity).
As an another example, our results also imply that if f is differentiable
almost everywhere, then Mf is differentiable a.e.
