I'm taking a quote from my textbook:

"We can think of the equation xy=1 as stating that two functions of x (namely, xy and 1) are equal. It follows that the derivatives of these functions are equal, so

$\displaystyle x\frac{dy}{dx}+y=0$"

This is stated very plainly and I am finding that I'm not following what is being said here. Maybe its the notation that I don't understand. I'm trying to figure out how xy=1 transformed into the equation set to zero. For some reason y is left as y. I'm seeing sometimes that y gets changed to $\displaystyle \frac{dy}{dx}$. I'm trying to follow my textbook the best I can. Can someone explain this to me?

Thanks in advance...