Hoping for some advice as to how to tackle this question from Numerical analysis. I assume some form of the three point formula is used, however the mid/end point formula seem to fail since there is no common $h$ between the given $x_i$ values.
Given Points $x_1,x_2,x_3$, for $y\in\mathbb{R}$, let $$L_i(y)= \prod\limits^{3}_{k=1;k\neq i}\frac{y-x_k}{x_i -x_k}.$$ Then if $|y-x_i| = O(h)$ for $i = 1,2,3$, then $$f(y) = \sum\limits^{2}_{i=0}L_i(y)f(x_i) + O(h^2).$$ Given that $$L_1'(y) = \frac{2y-x_2-x_3}{(x_1-x_2)(x_1-x_3)},$$ $$L_2'(y) = \frac{2y-x_1-x_3}{(x_2-x_1)(x_2-x_3)},$$ and $$L_3'(y) = \frac{2y-x_1-x_2}{(x_3-x_1)(x_3-x_2)}.$$ Estimate $f'(0.2)$ given $f(0) = 1.2, f(0.3) = 1.6$ and $f(0.5) = 2.0$.