The diff( f(x), x$n ) calling sequence uses a database of core differentiation formulas, sum representations for functions, full partial fraction expansions, and tools from the gfun package, to compute formulas for the nth (integer order) derivative of a given expression. To compute derivatives of fractional order see fracdiff.

You can enter the command diff/x$n using either the 1-D or 2-D calling sequence. For example, diff(cos(x), x$n) is equivalent to .

To compute formulas for the nth integral, specify -n for the order, for instance as in (diff(expr, x$(-n)) - see example at the end of this page.

The environment variable _EnvFallingNotation allows you to select how "x to the n falling" is represented: x^falling(n) := x(x-1)(x-2)...(x-n+1) can be represented by the pochhammer symbol, GAMMA notation, or factorial notation.  Each has some advantages. The default value is pochhammer.

Note: The diff implicitly assumes that n is an integer. Substitution of fractional values into the resulting formula will not compute fractional derivatives - for that purpose use fracdiff. Depending on the case, symbolic order differentiation can be a computationally expensive operation; uncomputed sums in the output are represented using Sum, not sum.

The expression is recursively examined for simple expressions.  A direct formula for monomials of the form C*(x-a)^p is used when such patterns are matched in the input.  Rational functions are converted to full partial fraction form.

When complicated terms are found in the input, a sequence of increasingly-powerful heuristics is tried: guessing a differential equation satisfied by the term, converting it to hypergeometric form, or converting it to Sum form by means of the built-in functional database.

Benghorbal, Mhenni, and Corless, Robert M. "The nth derivative." SIGSAM Bull (Communications in Computer Algebra). Vol. 36 No. 1, (2002): 10-14. http://doi.acm.org/10.1145/565145.565149