The Hilger derivative on a time scale is defined by
.
For discrete time,
and
The Hilger derivative becomes the first order difference for discrete time.
.
For continuous time, the Hilger derivative is taken in the limit sense and is the conventional continuous time derivative.
.
Integration on a time scale can be viewed as an antiderivative. If we have the time scale derivative
,
then the antiderivative is
,
and the definite integral on a time scale follows as
.
