given:

$\displaystyle xy^\frac{1}{2} + yx^\frac{1}{2} = 6$

we are trying to find $\displaystyle \frac{dy}{dx}$ so then, wouldn't you want to isolate y?

eventually you want a $\displaystyle m_2=\frac{1}{m_1}$

instead the book starts turning the original formula on this page into a derivative. I am in grants notes and have been stuck on this question for some time. If I work on it more I'll update this but if anyone thinks they can explain it, please do.