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. How many [math]\displaystyle{ (x, \ y) }[/math] pairs satisfy the previously mentioned equation? Infinitely many, it's an equation that in 2D describes some curve or path. In the same way we can plot functions point by point and connecting them with straight lines, a complicated curve can also be approximated with the same procedure.

The previous given curve could be rewritten as [math]\displaystyle{ [f(x)]^3 + x^2[f(x)]^2 + 4x = 0 }[/math] if we remember that [math]\displaystyle{ y = f(x) }[/math]. There is a function that we don't know, but the substitution made it clear that there is a relationship between the two variables.