let be a piecewise continous function or 'staircase' (the function is constant everywhere but has jumps) $ F(x)$
in the sense of distribution is always true that
$$ \frac{dF}{dx}= \sum_{n}\delta (x-a(n)) $$
where $ a(n) $ are the discontinuity points of F(x) where the function has jumps
for example $$ d[x] = \sum_{n=-\infty}^{\infty}\delta (x-n) $$
In general no, the derivative also depends on the "size" of the jumps. If $F$ is piecewise constant with discontinuities at $a_1, a_2, \ldots$, then
$$ F'(x) = \sum_j \big(\lim_{t\to a_j^+} F(t) - \lim_{t\to a_j^-} F(t) \big) \delta(x-a_j) $$