# Implementing discrete derivative by integration over a set of data

I'm trying to implement Lanczos derivative $$f'_L$$ as an option to calculate over a set of data. Since i needed a discrete version of Lanczos derivative $$f'_{DL}$$, i rearranged the equation such as $$f'_L(x) \approx \frac{3}{2h}\int_{-1}^{1}tf(x + ht)\,\mathrm{d}t = \frac{\int_{-1}^{1}tf(x + ht)\,\mathrm{d}t}{h\int_{-1}^{1}t^2\mathrm{d}t}.$$ By taking the quotient of Riemann sums as $$N\to\infty$$, it yields $$f'_{DL}(x) \approx \frac{N^{-1}\sum\limits_{t = -N}^{N}tf(x + ht/N)/N}{h\sum\limits_{t = -N}^{N}(t/N)^2} = \frac{3}{2N(N+1)(N+\frac{1}{2})}\frac{N}{h}\sum_{t = -N}^{N}tf(x + N^{-1}ht).$$ At this point, i should minimize $$N^{-1}h$$ in order to obtain a good estimate for the derivative, but since i'm limited to my data set, the minimum value will always be the next one on my list. The thing is, that this would mean that $$N^{-1}h = 1$$ (This being the next point). By taking the $$N=h=1$$, one immediately obtains the symmetrical derivative, making this process quite useless. So i wanted to know if there is any real value about implementing this derivative into a finite set of data.