Polynomial interpolation using derivatives at some points Given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)$, we can interpolate a polynomial of degree 4 using Lagrange method.  
But, when we are given $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y'''_5)$, how can we interpolate the same degree-4 polynomial?
Will Birkhoff interpolation be a good method to solve this or some modified version of Lagrange can be used?

One method, though I am unsure of the accuracy would be:
$y'''_5\approx \frac{y_5-3y_4-3y_3+y_2}{(x_5-x_4)(x_4-x_3)(x_3-x_2)}$ You can rearrange this to find an approximate $y_5$, and then you can use lagrange polynomials as normal, either using all data points for a degree $5$ approximation, or just use $4$ of them for a degree $4$ approximation.
