cubic, and quartic in , the same is not in general true for quintic and higher degree equations, such as Nevertheless, one can still refer to the implicit solution involving the multi-valued implicit function . Caveats Not every equation implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by where is a cubic polynomial having a "hump" in its graph. Thus, for an implicit function to be a true (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the -axis and "cutting away" some unwanted function branches. Then an equation expressing as an implicit function of the other variables can be written. The defining equation can also have other pathologies. For example, the equation does not imply a function giving solutions for at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling these sorts of pathologies. Implicit differentiation In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. To differentiate an implicit function , defined by an equation , it is not generally possible to solve it explicitly for and then differentiate. Instead, one can totally differentiate with respect to and and then solve the resulting linear equation for to explicitly get the derivative in terms of and . Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use. Examples Example 1 Consider This equation is easy to solve for , giving where the right side is the explicit form of the function . Differentiation then gives . Alternatively, one can totally differentiate the original equation: Solving for gives the same answer as obtained previously. Example 2 An example of an implicit function for which implicit differentiation is easier than using explicit differentiation is the function defined by the equation To differentiate this explicitly with respect to , one has first to get and then differentiate this function. This creates two derivatives: one for and another for . It is