Implicit differentiation is a mathematical technique used to find the derivative of a function that is defined implicitly by an equation, rather than explicitly. This means that the dependent variable is not explicitly written in terms of the independent variable.
Implicit differentiation is used when a function cannot be easily expressed in terms of one variable, making it difficult to use traditional differentiation methods. It allows us to find the derivative of a function without having to solve for the dependent variable first.
In explicit differentiation, the dependent variable is written explicitly in terms of the independent variable, making it easy to use traditional differentiation methods. In implicit differentiation, the dependent variable is not written explicitly, so we use the chain rule and other differentiation rules to find the derivative.
The steps for performing implicit differentiation are:
Implicit differentiation has many applications in physics, engineering, and economics. It can be used to find rates of change in complex systems, such as in thermodynamics and fluid mechanics. It can also be used to optimize functions in economics and to analyze curves in computer graphics and image processing.