I believe that the method used by ListPlot for interpolation is to interpolate each coordinate as a function of the list index. Something like the following looks a lot like the output from ListPlot[...,InterpolationOrder->3]:

With[{
  xyInterpolation=Interpolation[#,InterpolationOrder->3]&/@Transpose[myPoints]},
  ParametricPlot[Through[xyInterpolation[i]],{i,1,Length[myPoints]}]
]

From such an interpolation you should be able to grab your derivatives via implicit differentiation, e.g. dx/dy == (dx/dt)/(dy/dt). A delight to flaunt that notation in a place where it might make some mathematicians puke :)