Which of the two examples is more appropriate?
The derivative can be approximated with a finite-difference scheme.
The derivative can be approximated as a finite-difference scheme.
I believe that the most appropriate preposition to use is "with", given that "as" should intuitively be followed by a formula (at least, this sounds better to me).
For example
The derivative can be approximated as
f(x+1) - f(x)
.