File:Fractional Derivatives-Euler.gif

f = x^2;
df[\[Alpha]_] := Gamma[2 + 1]/Gamma[2 + 1 - \[Alpha]] (x)^(2 - \[Alpha]);
frames1 = Table[
   Plot[{Re[f], 2 x, 2, Re[df[\[Alpha]]], Im[df[\[Alpha]]]}, {x, -2, 2}, PlotRange -> {-4, 4}, PlotStyle -> {Directive[Black, Dashed], Directive[Gray, Dashed], Directive[LightGray, Dashed], Purple, Orange},     PlotLegends -> LineLegend[{"f=\!\(\*SuperscriptBox[\(x\), \(2\)]\)", "\!\(\*SubscriptBox[\(\[PartialD]\), \(1\)]\) f", "\!\(\*SubscriptBox[\(\[PartialD]\), \(2\)]\) f", StringForm["Re[\!*SubscriptBox[\(\[PartialD]\), \(``\)]\) f]", NumberForm[\[Alpha], {2, 1}]], StringForm["Im[\!\(\*SubscriptBox[\(\[PartialD]\), \(``\)]\) f]", NumberForm[\[Alpha], {2, 1}]]}], PlotLabel ->      "\!\(\*SubscriptBox[\(\[PartialD]\), \\(\[Alpha]\)]\)\!\(\*SuperscriptBox[\(x\), \\(n\)]\)=\!\(\*FractionBox[\(\[CapitalGamma] \((n + 1)\)\), \(\\[CapitalGamma] \((n + 1 - \\[Alpha])\)\)]\)!\(\*SuperscriptBox[\(x\), \(n - \[Alpha]\)]\)", LabelStyle -> {Bold, Black}], {\[Alpha], 0., 2., 0.02}];
ListAnimate[Join[Table[frames1[[1]], 4], frames1, Table[frames1[[-1]], 4], Reverse[frames1]]]