kædereglen

The chain rule is a fundamental rule in calculus used to differentiate composite functions. A composite function is a function within a function, often expressed as f(g(x)). The chain rule states that the derivative of a composite function f(g(x)) is the derivative of the outer function f, evaluated at the inner function g(x), multiplied by the derivative of the inner function g(x). Mathematically, if y = f(u) and u = g(x), then the derivative of y with respect to x, dy/dx, is given by dy/du * du/dx.

This rule is essential for differentiating more complex functions that cannot be directly differentiated using basic rules. For example, to differentiate (x^2 + 1)^3, we can consider the outer function f(u) = u^3 and the inner function g(x) = x^2 + 1. The derivative of the outer function is f'(u) = 3u^2, and the derivative of the inner function is g'(x) = 2x. Applying the chain rule, the derivative of (x^2 + 1)^3 is f'(g(x)) * g'(x) = 3(x^2 + 1)^2 * (2x), which simplifies to 6x(x^2 + 1)^2. The chain rule is a cornerstone of differential calculus and finds widespread application in physics, engineering, economics, and many other fields where rates of change are analyzed.