The aim of this work is to find closed form formulas that give the
nth derivative and the nth anti-derivative of elementary and special
functions. Here, we mainly concentrate on elementary functions
and give some theorems and techniques for finding the nth derivative
and the nth anti-derivative of integer orders. In general, n is a
symbol, but it can be replaced by a real number. We will be focusing
on the case when n is an integer.
The motivation of this work comes directly from the area of classical
and fractional calculus as well as the area of symbolic computation.
It is the answer to the question: Given a function f in a variable x,
can computer algebra systems (CAS) find a formula for the nth
derivative or the nth anti-derivative or both? A direct application
of the nth derivative formulas is in the area of classical
calculus. It is related to the construction of Taylor's series at
a point x0 where one requires the nthn derivative of a function
at the point where we approximate at. Other applications are related
to solving ordinary and fractional differential equations.
In Maple, the formulas correspond to invoking the command
diff(f(x), x$n) for differentiation. A software exhibition
will be within the talk.