compute a (partial) symbolic integer order derivative (or integral) of an expression
The Computational Approach
diff( f(x), x$n )
diff( f(x), x$(-n) )
algebraic expression depending on x to be differentiated (or integrated)
name; differentiation (or integration) variable
symbol understood to be an integer representing the differentiation (or integration) order
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 ⅆnⅆxncos⁡x.
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.
Compute the nth derivative of cos(x).
cn ≔ ⅆnⅆxn⁢cos⁡x
Compare with the result obtained by direct differentiation.
c3 ≔ ⅆ3ⅆx3⁢cos⁡x
Compute the nth integral of ⅇ2⁢x.
A basic formula: symbolic derivative of a monomial:
A more difficult function:
tn ≔ ⅆnⅆxn⁢arctan⁡x
Compute the formula for the nth derivative of sin(x).
Now compute the nth integral of the result.
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
Sum or sum
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