Implicit differentiation is a tool used to analyze functions that cannot be conveniently put into a form where . To use implicit differentiation meaningfully, you must be certain that your function is of the form (it can be written as a level set) and that it satisfies the implicit function theorem ( must be continuous, its first partial derivatives must be continuous, and the derivative with respect to the implicit function must be non-zero). To actually differentiate implicitly, we use the chain rule to differentiate the entire equation.

Example: The first step is to identify the implicit function. For simplicity in the example, we will assume and is an implicit function of . Let (Since this is a two dimensional equation, all one has to check is that the graph of may be an implicit function of in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to . We will get

[better example and ?multidimensional? coming]