I am supposed to derive the newtonian equation for gravitational acceleration ($-GM/r^2$) from the Schwarzschild metric $$ds=-(1-\frac{r_s}{r})c^2dt+(1+\frac{r_s}{r})^{-1}dr+0$$ where $$r_s=\frac{2GM}{c^2}$$ as we are only on a radial path (falling inward towards the planet) we are allowed to set $d\theta$ and $d\phi$ to $0$ so I didn't bother writing them. From here we are supposed to extremize $\int \frac{\sqrt{-ds}}{c}$ in order to somehow get acceleration = $$\frac{d^2r}{dt^2}=\frac{GM}{r^2}.$$ We are allowed to approximate $r_s<<r$ and $\frac{dr}{dt}<<c$ but I can't find a way to use either of these without totally breaking the equation
Attempt at a solution:
$$ds^2=-(1-\frac{r_s}{r})c^2dt+dr^2(1-\frac{r_s}{r})^{-1}\therefore\frac{\sqrt{-ds^2}}{c}=\sqrt{(1-\frac{r_s}{r})dt^2-dr^2((1-\frac{r_s}{r})c^2)^{-1}}=L$$ therefore $$L=dt\sqrt{(1-\frac{r_s}{r})-\frac{\dot{r}^2}{c^2-\frac{2MG}{r}}}$$
I should somehow use $$\frac{d}{dt}(\frac{dL}{d\dot{r}})-\frac{dL}{dr}=0$$ but the derivatives are so messy I don't know how to continue. I've seen other "solutions" like the Lagrangian approach on wikipedia but none seem to derive $GM/r^2$.