Keðjuregla

Keðjuregla, also known as the chain rule, is a fundamental principle in calculus used to find the derivative of a composite function. A composite function is one where the output of one function is used as the input for another. The chain rule provides a method to differentiate such functions efficiently.

The chain rule states that if y = f(u) and u = g(x), then the derivative of y with respect to x is given by the product of the derivative of f with respect to u and the derivative of g with respect to x. Mathematically, this is expressed as:

dy/dx = dy/du * du/dx

This rule is essential in various fields, including physics, engineering, and economics, where functions are often composed of multiple layers.

For example, consider the function y = (x^2 + 1)^3. To find the derivative of y with respect to x, we first identify the inner function u = x^2 + 1 and the outer function y = u^3. Applying the chain rule, we get:

dy/du = 3u^2

du/dx = 2x

Thus, the derivative of y with respect to x is:

dy/dx = 3u^2 * 2x = 6x(x^2 + 1)^2

The chain rule simplifies the differentiation process for composite functions, making it a crucial tool in calculus and its applications.