How are these two methods for numerical differentiation connected?

Question

I've read a book about Numerical Differentiation, and I found this formula: coefficient

derivative

I think this might be connected to the derivative of Lagrange Interpolating Polynomial, if I'm not mistaken they are equal. I just don't know how to prove it.

Answer 1

Use Lagrange-interpolation on $f(x)-f(x_i)$ to obtain $$ f(x)-f(x_i)=\sum_{j\ne i}(f(x_j)-f(x_i))\prod_{m\ne j}\frac{x-x_m}{x_j-x_m}+\prod_{j=0}^n(x-x_j)·r(x) $$ with $r$ some continuous function related to the $(n+2)$nd derivative.

Divide by $(x-x_i)$ $$ \frac{f(x)-f(x_i)}{x-x_i}=\sum_{j\ne i}\frac{f(x_j)-f(x_i)}{x_j-x_i}\prod_{m\ne i,j}\frac{x-x_m}{x_j-x_m}+\prod_{j\ne i}(x-x_j)·r(x) $$ Take the limit $x\to x_i$ $$ f'(x_i)=\sum_{j\ne i}\frac{f(x_j)-f(x_i)}{x_j-x_i}\prod_{m\ne i,j}\frac{x_i-x_m}{x_j-x_m}+\prod_{j\ne i}(x_i-x_j)·r(x_i) $$