Implicit Differentiation

Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. Unlike explicit functions, where the dependent variable is expressed explicitly in terms of the independent variable, implicitly defined functions involve an equation where the dependent variable and independent variable are intermingled.

Implicit differentiation relies on the chain rule to differentiate both sides of an implicit equation with respect to the independent variable. This process generates an equation involving the derivative of the dependent variable, which can then be solved for the derivative.

Implicit differentiation is useful when dealing with complex equations or functions that are difficult or impossible to solve explicitly for the dependent variable. Applications of implicit differentiation include finding tangent lines to curves, analyzing the behavior of multivariable functions, and solving differential equations.