  We define the notions of unilateral metric derivatives and ``metric derived
numbers'' in analogy with Dini derivatives (also referred to as ``derived
numbers'') and establish their basic properties. We also prove that the set of
points where a path with values in a metric space with continuous metric
derivative is not ``metrically differentiable'' (in a certain strong sense) is
$\sigma$-symmetrically porous and provide an example of a path for which this
set is uncountable. In the second part of this paper, we study the continuous
metric differentiability via a homeomorphic change of variable.
