fractional derivatives of xe^x

xex_frac_deriv.nb

(* found at https://community.wolfram.com/groups/-/m/t/1313893 *)
(* \
modified as per reddit user u/wxehtexw to integrate from -inf instead \
of 0 *)
FractionalD[\[Alpha]_, f_, x_, opts___] := 
 Integrate[(x - t)^(-\[Alpha] - 1) (f /. x -> t), {t, -Infinity, x}, 
   opts, GenerateConditions -> False]/Gamma[-\[Alpha]]
FractionalD[\[Alpha]_?Positive, f_, x_, opts___] := 
 Module[{m = Ceiling[\[Alpha]]}, 
  If[\[Alpha] \[Element] Integers, D[f, {x, \[Alpha]}], 
   D[FractionalD[-(m - \[Alpha]), f, x, opts], {x, m}]]]
   
(* sanity checks *)
f[x_] := x^2
FractionalD[1, f[x], x]
f[x_] := Sin[x]
FractionalD[-1, f[x], x]
f[x_] := Exp[x]
FractionalD[1, f[x], x]
FractionalD[1/2, f[x], x]
Simplify[FractionalD[3/2, f[x], x]]

(* test on xe^x *)
g[x_] := x Exp[x]
FullSimplify[FractionalD[1/4, g[x], x]]
FullSimplify[FractionalD[1/3, g[x], x]]
Simplify[FractionalD[1/2, g[x], x]]
FullSimplify[FractionalD[2/3, g[x], x]]
FullSimplify[FractionalD[3/4, g[x], x]]
Simplify[FractionalD[1, g[x], x]]
FullSimplify[FractionalD[5/4, g[x], x]]
FullSimplify[FractionalD[4/3, g[x], x]]
Simplify[FractionalD[3/2, g[x], x]]
Simplify[FractionalD[2, g[x], x]]