Show the gravitional field is conservative

[Physics2341313]

In my calculus textbook (section on vector calc) it is showing that the gravitational field is conservative. I followed fine except for the first part, defining the scalar function f.

Showing the field is conservative went something like this:

$f(x,y,z) = MM'G/\sqrt{x^2+y^2+z^2}$
$\nabla{f(x,y,z)} = \partial{f}/\partial{x}\hat{i}+\partial{f}/\partial{y}\hat{j}+\partial{f}/\partial{z}\hat{k}$
$= -MM'G/(x^2+y^2+z^2)^{3/2}\hat{i} + -MM'G/(x^2+y^2+z^2)^{3/2}\hat{j} + -MM'G/(x^2+y^2+z^2)^{3/2}\hat{k}$
= F(x, y, z)

Why, when defining the scalar function f is the $\sqrt{x^2 + y^2 + z^2}$ used?

 

[mfb]

It is the distance in cartesian coordinates.
Why the inverse distance and not its square or something else? Well, the inverse distance leads to the right force, other definitions do not.