compute a (partial) symbolic integer order derivative (or integral) of an expression - Maple Help

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diff/x$n - compute a (partial) symbolic integer order derivative (or integral) of an expression

Calling Sequence

diff( f(x), x$n )

ⅆnⅆxnfx

diff( f(x), x$(-n) )

 

Parameters

f(x)

-

algebraic expression depending on x to be differentiated (or integrated)

x

-

name; differentiation (or integration) variable

n

-

symbol understood to be an integer representing the differentiation (or integration) order

Description

• 

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 ⅆnⅆxncosx.

• 

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 Computational Approach

  

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.

Examples

Compute the nth derivative of cos(x).

cn:=ⅆnⅆxncosx

cn:=cosx+12nπ

(1)

Compare with the result obtained by direct differentiation.

c3:=ⅆ3ⅆx3cosx

c3:=sinx

(2)

c3cnn=3|c3cnn=3

0

(3)

Compute the nth integral of ⅇ2x.

ⅆnⅆxnⅇ2x

2nⅇ2x

(4)

A basic formula: symbolic derivative of a monomial:

diffxm,x$n

pochhammermn+1,nxmn

(5)

Use a different notation for the "falling" function:

_EnvFallingNotation:=Γ:

diffxm,x$n

Γm+1xmnΓmn+1

(6)

A more difficult function:

tn:=ⅆnⅆxnarctanx

tn:=122nMeijerG0,0,12,,0,12+12n,12n,x2x1n

(7)

normalexpandevalcsimplifyⅆ5ⅆx5arctanxtnn=5|ⅆ5ⅆx5arctanxtnn=5

0

(8)

Compute the formula for the nth derivative of sin(x).

Diffsinx,x$n

ⅆnⅆxnsinx

(9)

value

sinx+12nπ

(10)

Now compute the nth integral of the result.

diff,x$n

sinx

(11)

See Also

convert/fullparfrac, convert/parfrac, D, diff, eval, evalf, fdiff, fracdiff, int, Sum or sum, value

References

  

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


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