How do you find the gradient of $r = \sqrt{x^2 + y^2 + z^2}$ using finite differentiation?

I know if I find it normally, the gradient is the unit vector $\hat{r}$. But how do I use finite differentiation to find it?
$$
g\_i = \frac{f(x\_0 + h e\_i) − f(x\_0)}{h}, 
$$
where
$$
e\_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad 
e\_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad 
e\_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.
$$