**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 = x^{2} + xy^{3}. 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):

Differential Equations
Parametric Equations
Curves
Conic Sections

Questions? Comments? Queries? Criticisms? Feel free to /msg xerxes02