Derivative of an interpolated function

Remember the chain rule. You feed Interpolation with very contradictory information: The first derivative does not fit the parameterization of the curve.

This works better:

i = Interpolation[Table[{{2 t}, Sin[t], 1/2 Cos[t]}, {t, 0., 4., 0.01}]];
GraphicsRow[{
  Plot[i[t], {t, 0, 4}],
  Plot[i'[t], {t, 0, 4}]
  }]

Alternatively, you may use

i = Interpolation[Table[{{t}, Sin[t], Cos[t]}, {t, 0., 4., 0.01}]];

There is no error.

Given

f = Interpolation[Table[{{2 t}, Sin[t], Cos[t]}, {t, 0, 4, 0.01}]]

when you make the plot

Plot[f[t], {t, 0, 8}]

it looks like a nice smooth curve which should have a smooth derivative, but if you plot a small section of the domain, like this

Plot[f[t], {t, 0, .1}]

you see it is actually highly oscillatory, which explains your derivative plot.