Final answer:
An implicit function in multivariable calculus is defined by an equation relating two or more variables, without solving for one variable directly in terms of others. Examples include the equation of a circle. Differentiating such functions requires the technique of implicit differentiation.
Step-by-step explanation:
An implicit function in multivariable calculus is a function that is defined not explicitly by giving one variable directly in terms of another, such as f(x, y) = x2 + y2 = 1, but rather through an equation that relates two or more variables. In other words, the relationship between the variables is expressed without isolating one variable on one side of the equation.
For example, the equation of a circle, x2 + y2 = r2 does not explicitly solve for y in terms of x or x in terms of y, but it implies a relationship between x and y that defines y implicitly as a function of x (or x as a function of y) within a certain domain. Such equations are useful for describing complex relationships that may not be easily represented by a single function or where solving for one variable in terms of the others is not straightforward.
Differentiating implicit functions requires the use of the implicit differentiation technique, since we cannot isolate the dependent variable before taking the derivative. This technique involves differentiating both sides of the equation with respect to x and then solving for the derivative dy/dx.