In numerical differentiation, symmetric Newton’s quotient (a.k.a. symmetric difference quotient) is known to more effectively approximate the first derivative of given function , in comparison with the (first-order) divided difference. This statement can be summarized as the following proposition.
for sufficiently small and .
This sounds intuitively correct, since taking average of left and right divided differences at the point seem to smoothen the drastic slope change that can occasionally occur only one side. This is shown by somewhat technical manner.
Proof. Note that is bounded around since is assumed to be of class as a function on . By the Taylor series expression of and at , we get:
Subtracting the second row from the first, and divided by yealds:
Similarly we have the lower order infinitesimal equation:
By taking and , we can evaluate the equation only by the order of infinitesimal with regard to and being constant when is sufficiently small, which finish the proof ■