Data Set 1: This is the first data set, the graph of the function you will be taking the derivative is as follows:

Plot[(1-Tanh[2*(5-x)]^2)-(1-Tanh[2*(5+x)]^2),{x,-10,10}, PlotRange->All]

Here is the export command, change the path to wherever you want it to be saved to on your computer, then upload this data set to the cluster to have your program compute its derivative.

Export["C:\\Users\\kfrin\\Desktop\\Data1.dat",Table[{x,(1-Tanh[2*(5-x)]^2)-(1-Tanh[2*(5+x)]^2)},{x,-10,10,0.001}]]

DATA1= Table[{x,(1-Tanh[2*(5-x)]^2)-(1-Tanh[2*(5+x)]^2)},{x,-10,10,0.001}];

Length[DATA1]

Here we have Mathematica take the derivative, and we plot it, storing the plot in MDATAPlot variable.

D[(1-Tanh[2*(5-x)]^2)-(1-Tanh[2*(5+x)]^2),x]

FP[x_]:=4 ((Sech[2 (5-x)])^2) Tanh[2 (5-x)]+4 ((Sech[2 (5+x)])^2) Tanh[2 (5+x)];

FP[0]

MDATA1Plot = Plot[{4 ((Sech[2 (5-x)])^2) Tanh[2 (5-x)]+4 ((Sech[2 (5+x)])^2) Tanh[2 (5+x)]},{x,-10,10}, PlotRange->All, PlotStyle->{Red,Thickness[0.001]}]

You need to import your derivative data here, and then plot it with the derivative Mathematica gives.  Note: you will need to change the path in the Import command.  The picture below is NOT the derivative of the data set, its just a place holder.  The second graph below will show both graphs at the same time.

FDDATA1 = Import["C:\\Users\\kfrin\\Desktop\\Data1Diff.dat"];

FDDATA1Plot = ListPlot[FDDATA1, PlotStyle->{Blue, PointSize[0.005]}, PlotRange->All];

Show[MDATA1Plot ,FDDATA1Plot]

Now we want to graph the error in our finite difference derivative, the following command should do it (fingers crossed). Note current graph does not represent actual error, its just a place holder.

DiffData1  =Table[{FDDATA1[[k,1]],N[Abs[FDDATA1[[k,2]]-FP[FDDATA1[[k,1]]]]]},{k,1,20001}];

ListPlot[DiffData1, PlotRange-> All, PlotStyle->{Blue,PointSize[0.001]}]

Data Set 2: Here is one more data set, you are to perform the same analysis on this data.

Plot[Piecewise[{{2.0*(x^2)^(1/3),x<0},{1.5*x^(1/3),x>= 0}}],{x,-5,5}, PlotRange->All]

Export["C:\\Users\\kfrin\\Desktop\\Data2.dat",Table[{x,Piecewise[{{2.0*(x^2)^(1/3),x<0},{1.5*x^(1/3),x>= 0}}]},{x,-5,5,0.0005}]]

Data Set 3:  Apply the same process that you did for Data Set 1 and Data Set 2 to a function of your choosing, make it interesting!

Temp  =Table[{x,Piecewise[{{2.0*(x^2)^(1/3),x<0},{1.5*x^(1/3),x>= 0}}]},{x,-5,5,0.0005}];

Length[Temp]

The following definition of Temp2 removes the errors in the x-coordinates that we were seeing in the data file.  I just wrapped the x-coordinate in a NumberForm command, allowing 4 digits of decimal precision!

Temp2  =Table[{NumberForm[x,4],Piecewise[{{2.0*(x^2)^(1/3),x<0},{1.5*x^(1/3),x>= 0}}]},{x,-5.0,5.0,0.01}];

Export["C:\\Users\\kfrin\\Desktop\\Data2.dat",N[Temp2,10]]