Problems with implicit differentiation

In a problem where I need to use implicit diff. to find the slope of a line such as:

y^2 + x^2 = 9

2y (dy/dx) + 2x = 0

dy/dx = -y/x

Where does the (dy/dx) after the 2y (in the second part) come from? I've already differentiated y^2, x^2, and 9. Why isn't it just 2y + 2x = 0? I've looked all over, and in every problem an extra (dy/dx) or two just seem to pop up out of nowhere. Thanks!