An implicitly defined curve is a curve that is described by an equation involving two or more variables, rather than by an explicit function in terms of one variable. The equation defines the relationship between the variables but does not explicitly provide one variable as a function of the other. Implicitly defined curves can be analyzed using techniques like implicit differentiation to find properties such as tangent lines and slopes.
For example, the equation defines a circle with radius centered at the origin. This equation describes the relationship between and , but does not provide an explicit function for in terms of , or vice versa. To find properties such as the slope of the tangent line at a point on the circle, we can use implicit differentiation.