I am asked to show that the formula: $$ f'(x)\sim \sum_{i=0}^n A_i f(x_i) $$
which is derived from differentiating the interpolation polynomial is similar to that derived from checking/evaluating the order of accuracy formulae and similar to that derived through Taylor expansion. I have found the error in all these cases to be of the form: $$e(x)=C f^{(k)}(x) h^n$$ which means that the error becomes zero for any polynomial of degree less than or equal to $k-1$. Hence, for any such polynomial, the above formula could be rewritten thus: $$f'(x)=\sum_{i=0}^n A_i f(x_i)$$ Now, as the error in the Taylor expansion would also be zero for such polynomial, $f(x)$ could be accurately replaced with:
$$ \sum_{i=0}^{k-1} f^{(i)}(x_0) \frac{(x-x_0)^i}{i!} $$ Thus, the formula would hold and be similar. Is this attempt correct? Am I missing something? Would sincerely appreciate some advice.