Derivation

Consider the result of Theorem 10 with the linear polynomial fit to two data points:

We can take the derivative of both sides of this expression. First, we get

> P1:=x->((x-x[1])/(x[0]-x[1]))*f(x[0])+((x-x[0])/(x[1]-x[0]))*f(x[1]);

> x[1]:=x[0]+h:x[0]:=x0:dp1:=diff(P1(x),x);

where we have used the notation . The error term becomes:

> err:=(D@@2)(f)(xi(x))*(x-x[0])*(x-x[1])/2;

> diff(err,x);

This is not a very useful expression. Manipulating this error term is made difficult by the unknown quantity and the way it depends on . However, this part of the error term vanishes if we evaluate it at either or . The two formula obtained respectively are

,

.

The former formula is known as the forward-difference formula, the latter as the backward-difference formula. Both can be written in the first form if we allow .

If the second derivative of is bounded over the interval , i.e. , then the maximum error is given by .

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