Won't that just give you the gradient at point x_i, ie. d/dt(S)?

We are talking about frequency domain aliasing here,

which is when you generate partials that would be above the nyquist frequencyi, so they get reflected down into low frequencies. It is not directly related to the differential of the signal, though a high differential is often indicative of an aliasing problem. Typically you prevent audio aliasing by generating the waveform in a way so that it contains no partials above nyquist, or by generating it at a sufficiently high sample rate that there are none, then decimating down.