For the function $f(x)=\frac{1}{x}$, I am trying to calculate the Newton form of the interpolating polynomial for the points $x_0=2$, $x_1=3$ and $x_2=4$.
I understand that the Newton form of the interpolating polynomial is given by $P_2(x) = f(x_0)+f[x_0,x_1](x-x_0)+f[x_0,x_1, x_2](x-x_0)(x-x_1),$
where
$f[x_0,x_1] = \frac{f(x_1)-f(x_0)}{x_1-x_0}$
and
$f[x_0,x_1, x_2] = \frac{f(x_0)}{(x_0-x_1)(x_1-x_2)}+\frac{f(x_1)}{(x_1-x_2)(x_1-x_2)}+\frac{f(x_2)}{(x_2-x_0)(x_2-x_1)}$,
however when I plug in the given values and simplify I keep getting the wrong answer. Could someone please verify that my formulas are correct? Thank you in advance!