Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly written in terms of one variable. In other words, it is used to find the rate of change of a function when the dependent variable cannot be easily isolated.
Implicit differentiation is used when a function is written in implicit form, meaning that the dependent variable is not explicitly expressed in terms of the independent variable. This commonly occurs in equations involving multiple variables or when the function is defined implicitly.
Explicit differentiation is used to find the derivative of a function that is written explicitly in terms of one variable. Implicit differentiation is used when the function is not explicitly written in terms of one variable. Additionally, implicit differentiation often involves using the chain rule and product rule, whereas explicit differentiation typically only involves the power rule and basic derivatives.
The steps for performing implicit differentiation are as follows:
Implicit differentiation is used in many fields, such as physics, engineering, and economics, to analyze and model complex relationships between variables. For example, it can be used to find the rate of change of a variable in a physical system or to optimize production functions in economics. It is also commonly used in multivariable calculus to solve optimization problems.