Answer
$$\frac{dr}{ds}=\frac{1-2s-2r}{2s-1}$$
Work Step by Step
$$2rs-r-s+s^2=-3$$
To find $dr/ds$, we would use the methods of implicit differentiation.
Differentiate both sides of the equation with respect to $s$: $$2\Big(s\frac{dr}{ds}+r\frac{ds}{ds}\Big)-\frac{dr}{ds}-\frac{ds}{ds}+\frac{d(s^2)}{ds}=\frac{d(-3)}{ds}$$
$$2\Big(s\frac{dr}{ds}+r\Big)-\frac{dr}{ds}-1+2s=0$$
$$2s\frac{dr}{ds}+2r-\frac{dr}{ds}-1+2s=0$$
Next, separate the elements with $dr/ds$ and those without it into 2 sides of the equation: $$2s\frac{dr}{ds}-\frac{dr}{ds}=1-2s-2r$$
$$\frac{dr}{ds}(2s-1)=1-2s-2r$$
Then calculate for $dr/ds$: $$\frac{dr}{ds}=\frac{1-2s-2r}{2s-1}$$
