It's called "implicit differentiation".

Basically, you can navigate through this if you always use the chain rule, and remember that dx/dx = 1 (= x', when differentiating w.r.t. x)

So for

x^2 + x – y^2 = 0

(x^2)' + (x)' - (y^2)' = 0

2x*x' + 1*x' -2y*y' = 0

2x + 1 - 2y*y' = 0

2x + 1 = 2y*y'

(2x + 1)/2y = y'