First derivative of Lagrange polynomial

Question

Given the Lagrange basis polynomial as:

$L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $

is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?

Answer 1

By the "logarithmic derivative" method, $$\frac{L'_i(x)}{L_i(x)}=\sum_{m=0,\ m\neq i}^n\frac1{x-x_m}.$$