The diff( f(x), x$n ) calling sequence computes a formula for the nth (integer order) derivative of the expression f(x). To compute derivatives of fractional order see fracdiff.

The diff( f(x), x$(-n) ) calling sequence computes a formula for the nth integral of the expression f(x).

The symbolic derivative is computed using a database of core differentiation formulas, sum representations for functions, full partial fraction expansions, and tools from the gfun package.

You can enter the command for symbolic differentiation using either the 1-D or 2-D calling sequence. For example, diff(cos(x), x$n) is equivalent to $\frac{{\ⅆ}^n}{\ⅆx^n}\cos \left(x\right)$.

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 command 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.

$\mathrm{cn}≔\mathrm{diff}\left(\cos \left(x\right)\,\mathrm{`\$`}\left(x\,n\right)\right)$

$\mathrm{cn}≔\cos \left(x+\frac{n\mathrm{\pi }}{2}\right)$

$\mathrm{c3}≔\mathrm{diff}\left(\cos \left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)$

$\mathrm{c3}≔\sin \left(x\right)$

$\mathrm{eval}\left(\mathrm{c3}-\mathrm{cn}\,n=3\right)$

$0$

$\mathrm{diff}\left(\exp \left(2x\right)\,\mathrm{`\$`}\left(x\,-n\right)\right)$

${\ⅇ}^{2x}{2}^{-n}$

$\mathrm{diff}\left(x^m,\mathrm{`\$`}\left(x\,n\right)\right)$

$\mathrm{pochhammer}\left(m-n+1\,n\right)x^{m-n}$

$\mathrm{tn}≔\mathrm{diff}\left(\arctan \left(x\right)\,\mathrm{`\$`}\left(x\,n\right)\right)$

$\mathrm{tn}≔\frac{2^n\mathrm{MeijerG}\left(\left[\left[0\,0\,\frac{1}{2}\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[-\frac{1}{2}+\frac{n}{2}\,\frac{n}{2}\right]\right]\,x^2\right)x^{1-n}}{2}$

$\mathrm{normal}\left(\mathrm{expand}\left(\mathrm{evalc}\left(\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{diff}\left(\arctan \left(x\right)\,\mathrm{`\$`}\left(x\,5\right)\right)-\mathrm{tn}\,n=5\right)\right)\right)\right)\right)$

$0$

$\mathrm{Diff}\left(\sin \left(x\right),\mathrm{`\$`}\left(x\,n\right)\right)$

$\frac{{\ⅆ}^n}{\ⅆx^n}\sin \left(x\right)$

$\mathrm{value}\left(\right)$

$\sin \left(x+\frac{n\mathrm{\pi }}{2}\right)$

$\mathrm{diff}\left(,\mathrm{`\$`}\left(x\,-n\right)\right)$

$\sin \left(x\right)$

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