Implicit numerical differentiation is a technique used in mathematical modeling to estimate the derivative of a function at a given point. It involves using various numerical methods to approximate the derivative without having an explicit formula for the function.
Explicit numerical differentiation involves using a formula or algorithm to directly calculate the derivative at a point, while implicit numerical differentiation uses iterative methods to approximate the derivative without an explicit formula.
Implicit numerical differentiation can be more accurate and stable than explicit methods, especially when dealing with functions that have complex or discontinuous behavior. It also allows for the estimation of derivatives for functions that do not have an explicit formula.
Some common methods include the Newton-Raphson method, the secant method, and the Gauss-Newton method. These methods use a combination of function evaluations and approximations to iteratively estimate the derivative.
Implicit numerical differentiation is used in a variety of fields, such as physics, engineering, economics, and computer science. It can be used to analyze and optimize complex systems, solve differential equations, and estimate the sensitivity of a system to changes in its parameters.