File:Fractional Derivatives-RiemannLiouville.gif

n = 1; a = 0; b = 0; f = y^3 - 4 y;
frames2 = Table[
   sol1 = If[\[Alpha] == 0, Evaluate[1/2 D[f /. {y -> x}, {x, n}]], (+1)^Ceiling[\[Alpha]]/(2 Gamma[\[Alpha]]) D[FullSimplify@Integrate[f/(x - y)^(1 - \[Alpha]), {y, a, x}, Assumptions -> {x > 0}], {x, n}] ]; 
   sol2 = If[\[Alpha] == 0, Evaluate[1/2 D[f /. {y -> x}, {x, n}]], (-1)^Ceiling[\[Alpha]]/(2 Gamma[\[Alpha]]) D[FullSimplify@Integrate[f/(x - y)^(1 - \[Alpha]), {y, x, b}, Assumptions -> {x < 0}], {x, n}] ];
   Plot[{f /. {y -> x}, Evaluate[D[f /. {y -> x}, x]], Re[sol1 + sol2]}, {x, -3, 3}, PlotStyle -> {Directive[Black, Dashed], Directive[Gray, Dashed], Purple}, PlotRange -> {-10, 10}, PlotRange -> {-2, 2},     PlotLabel -> "\!\(\*SubscriptBox[\(\), \(0\)]\)\!\(\*SuperscriptBox[\(D\), \(\\[LeftCeiling]\[Alpha]\[RightCeiling] - \\[Alpha]\)]\)f(x)=\!\(\*FractionBox[\(1\), \\(2\)]\)[\!\(\*FractionBox[\(1\), (\[CapitalGamma] \((\[Alpha])\)\)]\\)\!\(\*SubscriptBox[\(\[PartialD]\), \(\[LeftCeiling]\[Alpha]\\[RightCeiling]\)]\)\!\(\*SuperscriptBox[SubscriptBox[\(\[Integral]\),\ \(0\)], \(x\)]\)\!\(\*FractionBox[\(f \((x)\)\), SuperscriptBox[\((x \- y)\), \(1 - \\[Alpha]\)]]\)dy+\!\(\*FractionBox[SuperscriptBox[\((\(-1\))\), \(\\[LeftCeiling]\[Alpha]\[RightCeiling]\)], \(\[CapitalGamma] \((\\[Alpha])\)\)]\)\!\(\*SubscriptBox[\(\[PartialD]\), \(\[LeftCeiling]\\[Alpha]\[RightCeiling]\)]\)\!\(\*SuperscriptBox[SubscriptBox[\(\\[Integral]\), \(x\)], \(0\)]\)\!\(\*FractionBox[\(f \((x)\)\), \SuperscriptBox[\((x - y)\), \(1 - \[Alpha]\)]]\)dy]", LabelStyle -> {Bold, Black}, PlotLegends -> LineLegend[{"f=\!\(\*SuperscriptBox[\(x\), \(3\)]\)-4x", "\!\(\*SubscriptBox[\(\[PartialD]\), \(1\)]\) f", StringForm[        "\!\(\*SubscriptBox[\(\), \(0\)]\)\!\(\*SuperscriptBox[\(D\), \\(``\)]\) f", NumberForm[\[LeftCeiling]\[Alpha]\[RightCeiling] - \[Alpha], \{2, 1}]]}]], {\[Alpha], 1, 0, -0.019}];
ListAnimate[Join[Table[frames1[[1]], 4], frames1, Table[frames1[[-1]], 4], Reverse[frames1]]]