Numerical Differentiation Using Lagrange Polynomials

Question

Here is a snapshot of Richard Burden's "Numerical Analysis" where he is discussing numerical differentiation using Lagrange polynomials with node points $x_0,x_1$ and $x_2$. I cannot fathom how he derived the formulas using the variable substitutions he speaks of. Could anyone please elaborate on that?

Answer 1

A way of thinking about it: replace $x_0$ everywhere with $y_0$ and then set $y_0=x_0-h$ in the second one and $x_0-2h$ in the third one. This makes the arguments on the left side of the equations become $x_0$ which is what you want to have happen.

The other thing that is happening at the same time is factoring out the $1/2$.

From a slightly higher level view, you're getting Taylor approximations for the derivative at the leftmost point, the middle point, and the rightmost point, respectively. The substitution is just used to rename the point of expansion as $x_0$ and then write the other two points relative to that.