Proof

Take to be Newton's divided difference form of the interpolating polynomial. then define a new function as .

Since for the interpolating polynomial , then .

Thus has distinct zero's in the interval and the Generalised Rolle's Theorem applies. Therefore, there exists a such that or .

The only contribution from the interpolating polynomial to the -th derivative comes from the term .

The -th derivative of this term is given by .

Therefore, or .