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
 .
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