
There is another way to obtain the results stated in “Finite Difference Approximations of Derivatives“.
Let denotes
and
respectively and,
.
We define
.
By Taylor’s expansion around ,
Substituting (1-1), (1-2) into (1),
.
That is,
It follows that

Fig. 1
Solving (1-3) for (see Fig. 1) yields
Therefore,
or,
Now, let
.
From
and
we have,
.
It leads to

Fig. 2
whose solution (see Fig. 2) is
.
Hence,
i.e.,