First derivative approximation using function values in equidistant points
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Given a function $f: [x_0,x_4] → Bbb R$ and equidistant points $x_0, x_1, x_2, x_3, x_4$ so that $h=x_{i+1} - x_i > 0$ . Normally I would do, $f'(x) approx dfrac{f(x+h)-f(x)}{h}$ , but here I want to find approximation for $f'(x_2)$ as linear combination of function values in these points as $f'(x_2) approx sum_{j=0}^4 alpha_j f(x_j)$ . There are no boundary conditions so I can't make a system of linear equations, at least I see no straight forward way. Are there any standard methods of how to determine alphas related to the finite elements method, so that the formula will be as precise as possible for polynomials of as high order as possible? I see that this is closely related to the five-point midpoint formula https://www3.nd.edu/~zxu2/acms40390F15/Lec-4.1.pdf if one se...