  We define analogues of higher derivatives for $F_q$-linear functions over the
field of formal Laurent series with coefficients in $F_q$. This results in a
formula for Taylor coefficients of a $F_q$-linear holomorphic function, a
definition of classes of $F_q$-linear smooth functions which are characterized
in terms of coefficients of their Fourier-Carlitz expansions. A Volkenborn-type
integration theory for $F_q$-linear functions is developed; in particular, an
integral representation of the Carlitz logarithm is obtained.
