Implicit differentiation

An explicitly defined function is of the form [math]\displaystyle{ y = f(x) }[/math], such as [math]\displaystyle{ f(x) = x^2 + 2 }[/math]. An implicitly defined function does not present the variables neatly to the right and to the left. For example: [math]\displaystyle{ x^2 + y^2 = 4 }[/math]. It's not an equation of a function, it describes a circle according to analytical geometry. However, it does "hide" a function in it. We can isolate [math]\displaystyle{ y }[/math] to obtain [math]\displaystyle{ y = \pm \sqrt{x^2 + 1} }[/math]. Due to the circle being symmetric in all directions we could have isolated [math]\displaystyle{ x }[/math] to obtain the same equation, except that we'd have swapped the variables.

For complicated equations such as [math]\displaystyle{ y^3 + x^2y^2 + 4x = 0 }[/math] we can try, but it's going to be almost impossible to properly isolate the variable and find the explicit function. A graphical way to interpret such equations is to think on them in terms of analytical geometry.