How do I plot the derivative of a set of experimental points?

Here is some sample data, hopefully of the sort you are looking for

a=Table[{i, i^2 - 3 i + 2 + 0.3 Random[]}, {i, -1, 3, 0.1}];
ListPlot[a]

will display the data.

This will create a derivable interpolation.

b = Interpolation[a, Method -> "Spline"];

This is its derivative

c = b';

Plot them together.

Plot[Evaluate[{b[x], c[x]}], {x, -1, 3}]

This is the data (taken from the previous example):

a = Table[{i, i^2 - 3 i + 2 + 0.3 Random[]}, {i, -1, 3, 0.1}];

This is derivatives calculated in each point except the first:

derA = Differences[a] /. {x_, y_} -> y/x;

here we combine it into the list of pairs:

lst=Transpose[{Drop[Transpose[a][[1]], 1], derA}];

Now we can plot it along with the initial list:

    ListLinePlot[{a, Transpose[{Drop[Transpose[a][[1]], 1], derA}]}, 
 PlotStyle -> {Blue, Red}]

That is what you get:

I know the question is quite old but maybe this might be helpful for someone.

I again follow the data set from above and show two methods of performing the task in question:

a = Table[{i, i^2 - 3 i + 2 + 0.3 Random[]}, {i, -1, 3, 0.1}];
ListPlot[a]

See image above, I'm not allowed to post three links due to not having >= 10 reputation.

b = Interpolation[a, Method -> "Spline"];
c = b';
d = Interpolation[a, Method -> "Hermite"];
e = d';
Plot[Evaluate[{b[x],c[x],e[x]}],{x,-1,3},PlotStyle->{Black,Blue,Red}]

Note how the two differnet interpolation methods differ!

The - in my oppinion - best way to go if you have knowledge about your model is the following:

bp = NonlinearModelFit[a,A x^2 + B x + F,{A,B,F},{x}];
Normal[bp]
bp["ParameterTable"]

0.995015 x^2 - 2.99342 x + 2.1193
    Estimate    Standard Error  t-Statistic  P-Value
A   0.995015    0.00994791      100.023      1.2327*10^-47
B   -2.99342    0.0225051       -133.01      2.51291*10^-52
F   2.1193      0.016774        126.345      1.76511*10^-51

cp = bp';
Plot[Evaluate[{bp[x],cp[x]}],{x,-1,3}]