# Function Repository Resource:

# FractionalD

Calculate a fractional derivative

Contributed by: Sami Yrjänheikki
 ResourceFunction["FractionalD"][f,{x,α}] gives the fractional derivative ∂αf/∂xα.

## Details and Options

If α is an integer, ResourceFunction["FractionalD"][f,{x, α}] is equivalent to [f,{x,α}].
The Cauchy method uses the formula .
The Caputo method uses the formula .

## Examples

### Basic Examples

Half-derivative of x2:

 In:= Out= Taking the half-derivative twice yields the usual first derivative:

 In:= Out= ### Scope

If α is an integer, the result is the normal derivative:

 In:= Out= Half-derivative of elementary functions:

 In:= Out= In:= Out= In:= Out= Half-derivative of trig functions:

 In:= Out= In:= Out= Half-derivative of inverse trig functions:

 In:= Out= In:= Out= A more exotic fractional derivative:

 In:= Out= ### Options

#### Method

Choosing between Cauchy and Caputo may result in performance differences:

 In:= Out= In:= Out= ### Possible Issues

FractionalD uses integration under the hood. Sometimes the integral cannot be computed:

 In:= Out= ### Neat Examples

The fractional derivative smoothly interpolates between the function and its derivative:

 In:= Out= 