# Implicit Differentiation

Now it's time to throw a monkey wrench into the works, curve ball style. What happens when we mix two variables together, on both sides of the equation? Why calculus? Why?! We're entering the Twilight Zone; we're traveling to another dimension. Once the room stops spinning, we'll move on. Good now? Good. We can handle this.

A function is a special kind of relation where each input has exactly one output. Often we have a rule or a formula that tells us how to reach the input to the output, such as

*y* = *x* + 5.

In such cases we say the function is defined **explicitly**. The dependent variable appears all by itself on one side of the equation, and the other side gives a recipe for making the dependent variable from the independent variable.

We say a function is defined **implicitly** if we have an equation in which the dependent variable doesn't have a side of the equation all to itself. The equation

5*x*^{2} – 4*y* = 7

is an implicit definition of *y*, since it doesn't actually say how to go from *x* to *y*. If we solve this equation for *y* we find the explicit definition

We can say that *y* is "defined implicitly" even if *y* isn't actually a function. For example, the equation

*x*^{2} + *y*^{2} = 1

describes a circle of radius 1. A circle isn't a function, since it fails the vertical line test. However, we would still say that this equation implicitly defines *y*.

For a total headache, we might even say "*y* is an implicitly defined function" even though *y* isn't a function. We probably won't worry about this distinction in class, but we wanted to point it out for the sake of completeness.