Fractional derivatives extend ordinary integer-order derivatives to noninteger orders. The Grünwald-Letnikov fractional derivative is defined using a limit of fractional-order backward differences.
For noninteger order 
and positive integer 
with 
, two other definitions use the Riemann-Liouville
operator of fractional integration. The
Riemann-Liouville fractional
derivative differentiates after fractional
integration,
 |
(1)
|
whereas the Caputo fractional derivative applies fractional integration after ordinary differentiation,
 |
(2)
|
For 
that is 
times continuously differentiable on ![[a,b]](/images/equations/FractionalDerivative/Inline6.svg)
and 
, their difference is
 |
(3)
|
where 
is the gamma function (Podlubny 1999, pp. 75-76).
In particular, the Caputo fractional derivative
of a constant function is zero,
while the Riemann-Liouville
fractional derivative of the constant function 
for 
is 
.
The semiderivative is a fractional derivative
of order 
.
The Riemann-Liouville fractional derivative with lower terminal 
is implemented in the Wolfram
Language as FractionalD[f,
{x, alpha}]. The Caputo
fractional derivative with the same lower terminal is implemented as CaputoD[f,
{x, alpha}].
A fractional integral can be defined similarly. The study of fractional derivatives and integrals is called fractional calculus.
