Create a function in Python that performs Hermite Interpolation using Divided Differences. Use it to construct polynomials for the

following data:

a$)$

x f$($x$)$ f$($x$)$

$8\mathrm{.}317\mathrm{.}564923\mathrm{.}116256$

$8\mathrm{.}618\mathrm{.}505153\mathrm{.}151762$

b$)$

x f$($x$)$ f$($x$)$

$0\mathrm{.}1-0\mathrm{.}620499583\mathrm{.}58502082$

$0\mathrm{.}2-0\mathrm{.}28398663\mathrm{.}14033271$

$0\mathrm{.}30\mathrm{.}006600952\mathrm{.}66668043$

$0\mathrm{.}40\mathrm{.}248424402\mathrm{.}16529366$

Problem $2$ Construct the Hermite Interpolating polynomials from the previous question by hand to check that the polynomials are the same.