Coefficients of numerical approximation of differentiation

Question

there is given
$y^\prime(\frac{a+b}{2}) = \alpha y(a) + \beta y(b) + \gamma y^{\prime \prime}(a) + \delta y^{\prime \prime}(b) \quad \quad (\dagger)$
we want to find $\alpha, \beta, \gamma, \delta$ such that ($\dagger$) is as accurate as possible for high degree polynomials and what is error when $|a-b| \rightarrow 0$.
I actually don't know where I should start from, any tip?

Answer 1

You have four adjustable parameters, so should expect to be able to be exact for up to third degree. Let $y=cx^3+dx^2+ex+f$ and plug it and its derivatives into the formula. You get one equation from the terms in each of $c,d,e,f$, so have four equations in four unknowns. There will be a lot of symmetry, so the system will be easy to solve.

It is even a little easier if you assume $a=-b$ so the center point is $0$. The result will be the same but some of the terms will disappear.