Find the derivative (dy/dx) of the curve equation x^2 -y^2 +y = 1.

Most of the differentiation problems require us to apply one of the well known rules, be it product rule, quotient rule or chain rule. But those problems have one thing in common:  explicite formula for y, be it y = ln(x) or y = sin(x)/(x² + 1).

In our example it's too difficult to isolate y, hence we will have to use implicit differentiation e.g. we will have to differentiate each term of the equation with respect to x.  Differentiating  (d/dx) yields,

d/dx [x²]  -   d/dx [y²] + d/dx [y] = d/dx [1]  =>

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

(dy/dx) (2y -1) =  2x   =>     (dy/dx) = 2x/(2y-1)

In our solution, we used the fact that the derivative of y²  with respect to x is equal to 2y(dy/dx).