derivative filter transfer function
This is not a standard 5-point derivative formula, the corresponding transfer function of which is

$$H(z)=\frac{1}{12}\left(-z^2+8z-8z^{-1}+z^{-2}\right)\tag{1}$$

The figure below shows the magnitude responses of an ideal differentiator (red), of the standard 5 point approximation given by (1) (green), and of the approximation in your question (blue):



As you can see, the non-standard approximation approximates the ideal differentiator only in a frequency region up to about $1/10$ of the Nyquist frequency (half the sampling frequency), whereas the standard approximation approximates the ideal differentiator quite well up to about a normalized frequency of $0.3$.

The differentiator in your question is a maximally flat lowpass differentiator. It avoids amplification of high-frequency noise. It achieves a maximally flat approximation of the ideal differentiator at $\omega=0$ and it has a given number of zero derivatives at $\omega=\pi$ (Nyquist), i.e. it is has a flat response at Nyquist, which causes high frequencies to be attenuated.

The design formula is derived in this paper. There is also a Matlab program available to design such maximally flat differentiators: link. Your specific filter is obtained by the command h=lowdiff(3,0);