keðireglan

The keðireglan, also known as the "chain rule," is a fundamental rule in differential calculus used to compute the derivative of a composite function. A composite function is formed by applying one function to the result of another function. If we have a function y = f(u) and another function u = g(x), then the composite function is y = f(g(x)). The keðireglan states that the derivative of this composite function with respect to x is the derivative of the outer function f with respect to its argument u, multiplied by the derivative of the inner function g with respect to x. Mathematically, this is expressed as dy/dx = dy/du * du/dx, or equivalently, d/dx [f(g(x))] = f'(g(x)) * g'(x). The rule essentially breaks down the differentiation of a complex function into a product of simpler derivatives. It is widely used in various fields of mathematics, physics, engineering, and economics for solving problems involving rates of change and optimization. Understanding the keðireglan is crucial for mastering calculus and its applications.