This work provides a rigorous mathematical analysis of the relationship between Hermite interpolation and function approximation, such that at a certain point Aq ∈ 𝔸q, where 𝔸q, is the set of all collections Aq := {αi}qi=1 of points αi  such that –1 ≤ αq < ⋯ < α1 ≤ 1, q ∈ N. Approximation polynomials (if possible) must interpolate the function and its derivatives, where at such points the function value matches not only the polynomial but also its derivatives. This kind of interpolation is known as Hermite interpolation also investigates whether the interpolation conditions can be extended to derivatives of the functionf. The accuracy of approximation does not just rely on the smoothness of f, but also the location points of interpolation, such that the points in Aq lie in the interior of the interval I = [–1, 1] or on the boundary of interval I, and so depend on the odd or even for highest derivatives for the f. With the points from Ys ∈ 𝕐s, where 𝕐s a sets for every collection  Ys := {yi}si=1 for the point yi, such that –1 < ys < ⋯