Is it correct to derivate Stirling formula with respect $x$ to obtain numericaly the value of the derivative of a function for a particular point?
Here is Stirling formula for the first five terms:
$P\left(x\right)=\:y_0+q\frac{\Delta y_{-1}+\Delta y_0}{2}+\frac{q^2}{2!}\Delta ^2y_{_{-1}}+\frac{q\left(q^2-1\right)}{3!}\frac{\Delta ^3y_{-2}+\Delta ^3y_{-1}}{2}+\frac{q^2\left(q^2-1\right)}{4!}\Delta ^4y_{-2}...$
with:
$q=\frac{x-x_0}{2}$