In general, pretty cool stuff.

There's no need to use Sympy, it's kind of overkill. If f(x) = 1/x^2 - a, then f'(x) = -2x^-3. You can find x_new easily:

x_new = x - f(x) / f'(x)

      = x - (1/x^2 - a) / (-2x^-3)

      = x + (1/2) (x^3) (1/x^2 - a)

      = x + (1/2) (x - ax^3)

      = (3x - ax^3) / 2

      = x(3 - ax^2) / 2