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.