I don't really get this method, it wasn't explained thoroughly in the book. It says:

"Differentiate term by term through the function, remembering that in differentiating functions of y you are differentiating a function of a function".

Then it gives an example:

Find $\displaystyle \frac{dy}{dx}$ of $\displaystyle x^2-y^2 +3x = 5y$

$\displaystyle 2x-2y\frac{dy}{dx}+3=5\frac{dy}{dx}$

$\displaystyle \frac{dy}{dx}=\frac{2x+3}{2y+5}$

I don't know why $\displaystyle \frac{d}{dx}(y^2)=2y\frac{dy}{dx}$. I know that y is really a polynomial in x and that 'somehow' complicates things, but it doesn't make sense to me to just add a $\displaystyle \frac{dy}{dx}$ to the end

I also never knew you could differentiate an 'equation'... could you effectively write this $\displaystyle \frac{d}{dx}($Equation$\displaystyle )$?