Rui,

The problem is that Derivative is a noisy operation. I improved things by making the numbers use infinite precision and converted them to 30 digits of precision before interpolation. Try Experimenting with this:

Manipulate[Nx = 2^pow; u0 = 0; uN = 5; du = (uN - u0)/Nx;
 ugrid = du Range[0, Nx];
 C0 = Exp[-(# - Lx/2)^2/2] & /@ ugrid; 
 exact[x] = D[Exp[-(x - Lx/2)^2/2], {x, der}];
 C0Poly = 
  Interpolation[N[Transpose[{ugrid, C0}], 30], 
   InterpolationOrder -> 10];
 exact[x] = D[Exp[-(x - Lx/2)^2/2], {x, der}];
 Plot[{Derivative[der][C0Poly][x], exact[x]}, {x, 0, Lx}, 
  PlotRange -> All, 
  ImageSize -> Large], {{pow, 9, 
   Dynamic["Points: " <> ToString[2^pow]]}, 4, 15, 
  1}, {{der, 9, Dynamic["Deriv: " <> ToString[der]]}, 1, 12, 1}, 
 ContinuousAction -> False, LabelStyle -> "Text"]

It will show you that the number of points matters and actually fewer points will be smoother. For some reason (which I have not yet figured out) the derivative is better if you use a power of two in the number of points. The number of intervals must interact with the method of interpolation. I hope this helps.

Regards,

Neil

I will venture a guess that the more points, the greater the likelihood that the degree ten (or so) local polynomial interpolants have spikes in small regions. This can worsen the noisiness noted by @NeilSinger.

Again, this is just a guess.