Implicit Differentiation

here's the blurb for Implicit Differentiation:

IMPLICIT DIFFERENTIATION

I got a d in calculus, failed trigonometry
cant factor vectored inegrals, and one plus one just seems like three
and i despise articulating all this useless calculation
cuz girl you know that all i need is your implicit differentiation.

The oscillation of a double sine curve's all i really wanna see
The integral of e to the x is where i wanna be
an i know one day we'll find ourselves in integration
cuz baby i just love your implicit differenciation

Implicit Differentiation

Definition: A process for finding dy/dx when y is implicitly defined as a function of x by an equation of the form ƒ(x,y) = 0.

What Does this Mean?

In other words, Implicit Differentiation is a method to find the derivative of a function when separation of x and y is not possible, or one is unable to put a function in the familiar "y=" or "ƒ(x)=" forms. Take, for example, the relation y = x² + xy³. Now say one needs to find dy/dx. By basic differential techniques, this is not possible. This is where Implicit Differentiation comes in.

That's Great, but How Does One Accomplish This?

Relying on our foreknowledge of algebra, calculus, and basic diffentiation (duh!), one looks at y as a function of x, or ƒ(x). This means the "y"s in the equation above can be replaced with ƒ(x). (Note: This is helpful for learning, but not necessary in the long run.) Now treating y as a function of x, differentiate as normal/take the implicit derivative:

     2    3
y = x + xy 

d      d   2    3                3       2 
-- y = -- x + xy   =  y' = 2x + y + 3xy'y     (Product Rule)
dx     dx

cleaning up yields

              3
        2x + y

y'  =  ---------
               2
        1 - 3xy  
         

Most of the time, this technique produces another implicitly defined function (relation). With values of x and/or y, we can then calculate dy/dx for this function. All this is helpful in a multitude of topics, including(but not limited to):