  Given an infinite, compact, monothetic group $G$ we study decompositions and
structure of unbounded derivations in a crossed product C$^*$-algebra
$C(G)\rtimes\Z$ obtained from a translation on $G$ by a generator of a dense
cyclic subgroup. We also study derivations in a Toeplitz extension of the
crossed product and the question whether unbounded derivations can be lifted
from one algebra to the other.
