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

Since for the interpolating polynomial [Maple Math] , then [Maple Math] .

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

The only contribution from the interpolating polynomial to the [Maple Math] -th derivative comes from the term [Maple Math] .

The [Maple Math] -th derivative of this term is given by [Maple Math] .

Therefore, [Maple Math] or [Maple Math] .