We want to construct a piecewise cubic function that agrees with a function and its derivatives at the nodes.
This is the basis for Bezier curves.

For simplicity, assume we have x = f(t), where t are integers.

f[t_] := Sin[2.0  t];
tMax = 5;
xList = Table[f[t], {t, 0, tMax}];
vList = Table[f'[t], {t,0, tMax}];
xPlot = Table[Point[{t, f[t] }], {t, 0, tMax}];
vPlot =  Table[Line[{{t-1/3, f[t] - 1/3 f'[t]}, {t+1/3,  f[t] + 1/3 f'[t]}}], {t, 0, tMax}];
Graphics[{Cyan,vPlot, Black, xPlot}, Axes -> True, AxesLabel -> {t, x}]