# Should I use "with" or "as" in place of X in "The derivative can be approximated X a finite-difference scheme"?

Which of the two examples is more appropriate?

1. The derivative can be approximated with a finite-difference scheme.

2. 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)`.

In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations.

In your second example, I think "by" would be better.

The derivative can be approximated by f(x+1) - f(x).

If you are using a finite-difference scheme to create the approximation, but the finite-difference scheme is not itself the actual approximation, use "with".

If some particular finite-difference scheme is the approximation, use "as".

I think you understand this issue well. Both of these are frequently and interchangebly used in this case, but using with is a little more vague/general. The writer is saying that there exists a finite difference scheme that reasonably approximates the derivative, but may not actually specify which one, presumably because that's not the point.

On the other hand, it would be more likely to say as if you were about to actually insert the finite difference scheme in place of the derivative in your formula, or if you were about to put a comma and the formula itself.