Hermite interpolating polynomial ($H(x_i)=y_i,H^{\prime }(x_i)=d_i$)

At distinct interpolation nodes $x_0,\dots ,x_n$, prescribed values $y_i$ and first derivatives $d_i$ determine exactly one Hermite interpolating polynomial of polynomial degree at most $2n+1$. The homogeneous data force a double root of a polynomial at every interpolation node. Hence the kernel of a linear map given by these $2n+2$ conditions is zero: a nonzero polynomial with those double roots of a polynomial would have polynomial degree at least $2n+2$. Equal finite dimensions of a vector space then give existence as well as uniqueness. This interpolation polynomial is distinct from the Hermite polynomials forming a classical orthogonal polynomial family.