derivatives - Is there a difference in domain between the two ways to notate the reciprocal function?

Whilst exploring basic derivation, I noticed something peculiar.

We know that the derivative of any constant is 0. However, does this derivative exist for the whole domain?

Consider the function $f(x)=x^2$. We know $f'(x)=2x$, and $f''(x)=0$.

If you write the functions literally, it would come out to this:

$f(x)=x^2$

$f'(x)=2*x^1=2x$

$f''(x)=1*2x^0=2$

$f'''(x)=0*2x^{-1}=0$

The issue sprouts from this last derivative; since there is an $x^{-1}$, then wouldn't the derivative not exist if $x=0$? If it does exist, this brings up the question if this is mathematical syntax that allows it to exist. Another example is the following two functions, which are equivalent mathematically, but not by function.

$f(x)=\sqrt{\frac{x^3}{x}}$ Domain: ($-\infty, 0) U (0, \infty$)

$g(x)=\frac{\sqrt{x^3}}{\sqrt{x}}$ Domain: ($0,\infty$)

These two functions are taught to be mathematically equivalent, but do not yield the same domain. This is why the question of mathematical syntax has come up. Do these two functions yield the same domain?

$f(x)=1/x$

$g(x)=x^{-1}$