Geodesic radius of curvature

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula.

$\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$

where $s$ is the arc length parameter and $E$, $G$ are the coefficents of the first fundamental form.

Can you please tell me how to perfrom the $\partial G/\partial S$? Since $G=r_v\cdot r_v$ I am not sure how to derivate it with respect to arc length

Thanks!

Geodesic (or tangent) curvature of geodesic "parallel" circles in geodesic polar coordinates is given ( for $s = u$= constant lines, not $v$= constant ) by:
$1/ρ_c= \kappa_c=∂G/∂s/(2G).$
where s is the arc length parameter and G is the coefficient of the first fundamental form. You must choose your particular case parametrization next. For example in case of axis-symmetric cylindrical coordinates $G = r^2$ , where $d$ can be written for $∂$, constant Gauss curvature differential equation
$∂^2\sqrt G/∂s^2 + K \sqrt G=0$ becomes $d^2r/ds^2 + K r=0$