FractionalD
✖
FractionalD
gives the Riemann–Liouville fractional derivative of order α of the function f.
Details and Options

- FractionalD is also known as the Riemann–Liouville differintegral of f.
- FractionalD generalizes D to fractional order and unifies the notions of derivatives and integrals from calculus.
- FractionalD plays a foundational role in fractional calculus since other types of fractional derivatives such as CaputoD can be defined in terms of it.
- The Riemann–Liouville fractional derivative of
of order
is defined as
, where
.
- The derivatives of fractional order "interpolate" between the derivatives of integer orders, as shown below for the function
and its fractional derivatives of order
given by
for
:
- The order α of a fractional derivative can be symbolic or an arbitrary real number.
- FractionalD[array,{x,α}] threads FractionalD over each element of array.
- FractionalD takes different Assumptions on the parameters of input functions.
- All expressions that do not explicitly depend on the given variable
are interpreted as constants.

Examples
open allclose allBasic Examples (4)Summary of the most common use cases
Calculate the half-order fractional derivative of a quadratic function with respect to x:

https://wolfram.com/xid/0cg37llh7-9ijqxw

Arbitrary-order fractional derivative of a quadratic function with respect to x:

https://wolfram.com/xid/0cg37llh7-06lozd

Plot these fractional derivatives for different 's:

https://wolfram.com/xid/0cg37llh7-pggxbv

Calculate the -order fractional derivative of a constant with respect to x:

https://wolfram.com/xid/0cg37llh7-y2w9t3

Fractional derivative of MittagLefflerE:

https://wolfram.com/xid/0cg37llh7-gl9r59

Scope (8)Survey of the scope of standard use cases
Fractional derivative of the power function with respect to x:

https://wolfram.com/xid/0cg37llh7-zamx21

0.23-order fractional derivative of the Exp function with respect to x:

https://wolfram.com/xid/0cg37llh7-gvkdq9

For positive integer , the fractional Riemann–Liouville derivative coincides with the ordinary derivative:

https://wolfram.com/xid/0cg37llh7-b0uzfd


https://wolfram.com/xid/0cg37llh7-lfctso

For negative integer , FractionalD differs from the ordinary indefinite integral by a constant:

https://wolfram.com/xid/0cg37llh7-mdyb9p


https://wolfram.com/xid/0cg37llh7-l7c047


https://wolfram.com/xid/0cg37llh7-dv7zz3

Fractional derivatives of Sin function are written in terms of HypergeometricPFQ:

https://wolfram.com/xid/0cg37llh7-yyvdki

Fractional derivatives of BesselJ function:

https://wolfram.com/xid/0cg37llh7-e3yxgu

Fractional derivatives of MeijerG function are given in terms of another MeijerG function:

https://wolfram.com/xid/0cg37llh7-4k5kwb

Laplace transform of a fractional integral in general form:

https://wolfram.com/xid/0cg37llh7-o0d9hj

Substitute the exponential function:

https://wolfram.com/xid/0cg37llh7-en3a4l

Get the same result by applying LaplaceTransform to the FractionalD of Exp:

https://wolfram.com/xid/0cg37llh7-7l8v0j

Options (1)Common values & functionality for each option
Assumptions (1)
FractionalD may return a ConditionalExpression:

https://wolfram.com/xid/0cg37llh7-bais0b

Restricting parameters using Assumptions will simplify the output:

https://wolfram.com/xid/0cg37llh7-mu7kqu

Applications (2)Sample problems that can be solved with this function
Calculate the half-order fractional derivative of the cubic function:

https://wolfram.com/xid/0cg37llh7-il7axk

Get the ordinary derivative of the cubic function repeating the half-order fractional differentiation:

https://wolfram.com/xid/0cg37llh7-2aq2lc

Recover the initial function using fractional integration:

https://wolfram.com/xid/0cg37llh7-t34o6x

Consider the following fractional order integral equation:

https://wolfram.com/xid/0cg37llh7-xlagoz
Solve it for the Laplace transform:

https://wolfram.com/xid/0cg37llh7-i1d1r4


https://wolfram.com/xid/0cg37llh7-km5asc


https://wolfram.com/xid/0cg37llh7-1p7fu3

Properties & Relations (6)Properties of the function, and connections to other functions
FractionalD is defined for all real :

https://wolfram.com/xid/0cg37llh7-z2ojjm


https://wolfram.com/xid/0cg37llh7-n1trsx

The 0-order fractional derivative of a function is the function itself:

https://wolfram.com/xid/0cg37llh7-vm2in3

FractionalD is not defined for complex order :

https://wolfram.com/xid/0cg37llh7-ri9o7g


In general, the fractional derivative of a constant is not 0:

https://wolfram.com/xid/0cg37llh7-heuuox

FractionalD results may contain DifferenceRoot sequences:

https://wolfram.com/xid/0cg37llh7-fgc8rk

This general expression is simplified to a finite sum of HypergeometricPFQ instances if is a given real number:

https://wolfram.com/xid/0cg37llh7-0lugr2

Calculate the fractional derivative of a function at some point:

https://wolfram.com/xid/0cg37llh7-rpoch9

Use the NFractionalD function for faster numerical calculations:

https://wolfram.com/xid/0cg37llh7-s892kc

Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.
Text
Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.
Wolfram Research (2022), FractionalD, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalD.html.
CMS
Wolfram Language. 2022. "FractionalD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalD.html.
Wolfram Language. 2022. "FractionalD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalD.html.
APA
Wolfram Language. (2022). FractionalD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalD.html
Wolfram Language. (2022). FractionalD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalD.html
BibTeX
@misc{reference.wolfram_2025_fractionald, author="Wolfram Research", title="{FractionalD}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/FractionalD.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_fractionald, organization={Wolfram Research}, title={FractionalD}, year={2022}, url={https://reference.wolfram.com/language/ref/FractionalD.html}, note=[Accessed: 10-July-2025
]}