Let's say we have a set of coordinates in spherical co-ordinate system ,the coordinates are (r,θ,φ). To change the Spherical coordinates to Cartesian coordiantes we write it as
$$x=r\sin θ \sin φ, y=\sin θ \cos φ, z=r \cos θ$$
the position vector is given by ,
$$\vec ρ=r \sin θ \sin φ \vec i+r \sin θ \cos φ \vec j+r \cos θ \vec k$$
now to get the basis vector say $\vec θ$ we divide the derivative of position vector with respect to θ and the magnitude of the position vector with respect to $θ$,
that is
$$\vec θ =\frac{\frac{\partial \vec ρ}{\partial θ}} {\left|\frac{\partial \vec ρ}{\partial θ}\right|}$$
but why does a derivative divided by its magnitude give us the basis vector?
By definition multiplication of a vector by a scalar modify the “length” of the vector but not its direction that is
$$ \vec v=| \vec v|\frac{\vec v}{| \vec v|}$$
with $\frac{\vec v}{| \vec v|}$ unitary direction vector for $\vec v$.
Let try with some numerical example as $\vec v=(1,2,3)$.
Refer also to the related