# Chapter 3: Derivatives - Practice Exercises - Page 178: 52

$\frac{dr}{ds}=\frac{1-2r-2s}{2s-1}$

#### Work Step by Step

Take the derivative of the equation on each side separately. Apply chain rule when differentiating the "r" variables since we are differentiating with respect to s: $2r\times1+2s\frac{dr}{ds}-\frac{dr}{ds}-1+2s=0$ $2r+2s\frac{dr}{ds}-\frac{dr}{ds}-1+2s=0$ Move all terms with dr/ds to one side of the equation, and isolate dr/ds: $2s\frac{dr}{ds}-\frac{dr}{ds}=1-2r-2s$ $\frac{dr}{ds}(2s-1)=1-2r-2s$ $\frac{dr}{ds}=\frac{1-2r-2s}{2s-1}$

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