Derivation

Assume a method that would take the form

;

;

>

In order to obtain a local truncation error of order 2, this evaluation must match the term in Taylor's method of order 2 ( ) except for terms of order , with

.

The Taylor series expansion of the assumed evaluation yields:

.

Writing out in more detail yields:

Equating these two expressions term by term gives:

;

;

;

Therefore,

,

The error term is given by

,

which, if all derivatives are bounded, is of the order .

>

Therefore, rather than calculating the derivative of one can obtain a result with similar accuracy by evaluating at the point ( ).