Applying derivative and integral on noisy data

Let's say you have a motion with sinusoidal acceleration, i.e. xddot(t) = sin(t). The analytic solution for speed and position will be xdot(t) = -cos(t)+1 (assuming zero initial speed) and x(t) = -sin(t) + t.  Let's say you don't know the analytical expressions of position, speed and acceleration and only have their values at a fixed sample rate. If you want to increase the samples, you can use Matlab's spline function.

If you have position as input and calculate speed and acceleration, if there is any noise in the signal, it will be amplified. If you have acceleration as input and calculate speed and position via integration, noise will be attenuated. Another advantage of using integration instead of derivation is that we can use linear interpolation since when integrating, we don't need smooth second derivatives. This both decreases computational load and lag due to two points in linear interpolation compared to four points in spline. See below Matlab code for an example: