plots = Table[
   Plot[
    Sum[an[[j + 1]] Gamma[j + 1]/Gamma[j + 1 - \[Alpha]] (x)^(
      j - \[Alpha]), {j, Ceiling[\[Alpha]], nn}]
    , {x, 0.2, 3.8}, PlotRange -> {-1.5, 1.5}, Ticks -> None, 
    Axes -> {True, False}, PlotStyle -> {Thick, Blue}, 
    Epilog -> {Text[
       StringForm[
        "\!\(\*FractionBox[\(\*SuperscriptBox[\(\[PartialD]\), \
\(``\)]\*SuperscriptBox[\(e\), \(-\*SuperscriptBox[\(x\), \
\(2\)]\)]\), \(\[PartialD]\*SuperscriptBox[\(x\), \(``\)]\)]\)", \
\[Alpha], \[Alpha]], {0.5, 1}]}]
   , {\[Alpha], 0, 1, 0.01}];
Export["<path>/fractionalderivative.gif", plots]

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This file, which was originally posted to https://twitter.com/j_bertolotti/status/1030403517641568256, was reviewed on 19 October 2018 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.