Let us say that we have a discrete signal $I_n$, $n=0, 1, 2, ...$. According to Nyquist theorem the maximum frequency for such discretization is $f_{max} = 0.5$.
Now imagine that I want to calculate the derivative $D_n$ for this signal. The simplest approximation is the right hand side derivative
$$D_n = I_{n+1} - I_n$$
But what if I want to use a central derivative instead?
$$D_n = \frac{1}{2}(I_{n+1} - I_{n-1})$$
Just looking at this relation, it seems like I am increasing the sampling time from $1$ to $2$. Does this mean that before calculating central derivative I should apply low-pass filter to kill all the frequencies $f > 0.25$? Or am I being wrong and the simple fact that I am using the spacing = 2 for calculating derivative does not mean that I increase the sampling distance, because I can still calculate this derivative for each value of $n$?
No matter what the correct answer is (yes or no), can you please explain in more detail why? If the low-pass filter should be applied, can you explain if there is a method, how given discrete time relation I can infer to what degree I should smoothen the signal before using such relation?