Computers can't deal with limit of $\Delta x \to 0$

Question

While I was studying about finite differences I came across an article that says "computers can't deal with limit of $\Delta x \to 0$ " in finite differences.But if computers can't deal with these equations does anybody know how they compute $ \frac {d}{dx}$ of $x^2$ and other such equations.Or whether these equations are pre-written.

Answer 1

A table of the derivatives of primitive functions combined with differentiation rules yields an algorithm that allows a computer program to symbolically compute the derivative of any function that is a compound of primitive functions without having to rely on the limit definition at all (computer algebra).

In fact, unlike symbolic integration, this algorithm is fairly easy to implement.

Edit: It can only be re-emphasised that symbolic integration is a different matter entirely. In fact, while every function that is comprised of elementary functions has an elementary derivative, the same is not true for the antiderivative. This is a consequence of Liouville's theorem and finds application in the Risch "algorithm" (which is not an algorithm in the strict sense).

Answer 2

This is not an answer but an illustration of automatic differentiation of a formula (I used Tapenade software online).

The source code I submitted is

  SUBROUTINE DUMMY(X,Y)
    Y = X ** 2
  END

which was interpreted as

  SUBROUTINE DUMMY(x, y)
  IMPLICIT NONE
  REAL x , y
  y = x**2
  END

and what I received is

  SUBROUTINE DUMMY_D(x, xd, y, yd)
  IMPLICIT NONE
  REAL x , xd , y , yd
  yd = 2*x*xd
  y = x**2
  END

which now computes both the function $y$ and its derivative $\frac{dy}{dx}$ ($xd=x$).