How to Isolate all terms with $dy$ divided by $dx$ as a factor on one side of the equation

Question

The question i cant figure out is: $$6x^2+8xy+4y^2+17y-6=0$$ I understand that you are supposed to take the derivative but after i do it says to Isolate all terms with $dy$ divided by $dx$ as a factor on one side of the equation. I cant understand why some terms are $dy/dx$ and other are not.

Answer 1

When you take the derivative all variables other than the one that you are taking the derivative with respect to are assumed to be functions of that variable. So $$\frac{d}{dx}x^2=2x\quad\text{ and }\quad \frac{d}{dx}y^2=\frac{d(y^2)}{dx}\frac{dy}{dy}=\frac{dy}{dx}\frac{d(y^2)}{dy}=2y\frac{dy}{dx}$$

This is the result of chain rule for differentiation. $$\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}$$ So to answer your question the $dy/dx$ comes from the $y$ containing terms in the equation.