Thread: Hermite interpolation starting with a Lagrange form

Is there a simple way to derive a Hermite interpolation function starting from the Lagrange form.
e.g. Suppose we know and we want to interpolate between and .

I thought of setting up the Lagrange form for x_0, x_1, x_2, x_3, and letting x_2 -> x_0 and x_3 -> x_1, but that doesn't seem to work out.

Any help?