Even with @MattL.'s fix you are discarding typically non-zero parts of the discrete-time derivative by not including its first and last sample, which destroys its autocorrelation properties near the end points, typically resulting in the low-frequency plateau in the frequency spectrum as you have observed. We can add a bit of a zero-valued safety buffer at ...
When I try this, the results looks as expected. So you really have to explain in more detail what exactly it is that you're doing, because it doesn't seem to be a property of the discrete-time derivative.
EDIT: Now that I see your code, I'm convinced that the problem will disappear if you use
dydt = zeros(1,sr);
to initialize the derivative vector.
Your FFT method of finding the phase of a single tone is only valid if your 's' signal contains an exact integer number of cycles. I.E., no spectral leakage. And that is NOT the case for your 's' signal.
If I compute the Fourier Transform of Z(f), what will I get?
If you apply the inverse Fourier Transform, you get the impulse response of the system which is indeed a time domain signal. You can also apply the forward transform and would get the impulse response time reversed and scaled since the forward and inverse transform are quite similar.
Which is ...
Equidistant Dirac impulses in the spectrum imply a periodic time domain signal.
As pointed out in a comment, in continuous time, the signal $x(t)=\sin(\omega_0t)$ is always periodic, regardless of the value of $\omega_0$.
Your question about the spectrum being periodic is unclear to me.
A one-sided spectrum with equidistant Dirac impulses also implies a ...