For this one, when I try differentiate y(x) I end up with dy/dx = (2xy)(dy/dx) + y^2 instead of dy/dx = 2xy (I'm assuming y(x) and y mean virtually the same thing?)

For this one I tried to verify that this example was correct, but I found that dM/dy = 2(dx/dy) + cosy + (e^-x)(y)(dx/dy)-e^-x

and dN/dx = (-xsiny)(dy/dx) + cosy - (siny)(dy/dx) - e^-x.

So it seems that the terms with (dy/dx) or (dx/dy) automatically become zeroes or just don't matter, but I don't understand why?