substantially easier to implicitly differentiate the original equation: giving Example 3 Often, it is difficult or impossible to solve explicitly for , and implicit differentiation is the only feasible method of differentiation. An example is the equation It is impossible to algebraically express explicitly as a function of , and therefore one cannot find by explicit differentiation. Using the implicit method, can be obtained by differentiating the equation to obtain where . Factoring out shows that which yields the result which is defined for General formula for derivative of implicit function If , the derivative of the implicit function is given by where and indicate the partial derivatives of with respect to and . The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to — of both sides of : hence which, when solved for , gives the expression above. Implicit function theorem Let be a differentiable function of two variables, and be a pair of real numbers such that . If , then defines an implicit function that is differentiable in some small enough neighbourhood of ; in other words, there is a differentiable function that is defined and differentiable in some neighbourhood of , such that for in this neighbourhood. The condition means that is a regular point of the implicit curve of implicit equation where the tangent is not vertical. In a less technical language, implicit functions exist and can be differentiated, if the curve has a non-vertical tangent. In algebraic geometry Consider a relation of the form , where is a multivariable polynomial. The set of the values of the variables that satisfy this relation is called an implicit curve if and an implicit surface if . The implicit equations are the basis of algebraic geometry, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials. These sets of simultaneous solutions are called affine algebraic sets. In differential equations The solutions of differential equations generally appear expressed by an implicit function. Applications in economics Marginal rate of substitution In economics, when the level set is an indifference curve for the quantities and consumed of two goods, the absolute value of the implicit derivative is interpreted as the marginal rate of substitution of the two goods: how much more of one must receive in order to be indifferent to a