The implicit differentiation method states:

An equation f(x,y) = 0 defines y implicitly as a function of x. In order to to find the derivative of y, differentiate both sides of the original equation f(x, y) = 0 and solve the resulting equation for dy/dx. This differentiation method is known as implicit differentiation.

For example, given x³ + y³ - 4xy = 0, find dy/dx.

Differentiate both side of x³ + y³ - 4xy = 0 with respect to x, gives

                  (1)

Since y is implicitly defined by x, d(y³)/dx is not 0. Consider z = y³ and apply Chain Rule,

                                        (2)

Recall that if F(x) = f(x)g(x), then F '(x) = f '(x)g(x) + f (x)g '(x).The formula can be used to calculate d(4xy)/dx. Consider f(x) = x and g(x) = y, thus

                        (3)

Substitute equation(2) and (3) into (1),

so

     3x² + 3y²dy/dx - (4y + 4xdy/dx) = 0

     (3y² - 4x)dy/dx = 4y - 3x²

     dy/dx = (4y - 3x²)/(3y² - 4x)