Kettenregel

The Kettenregel, or chain rule, is a fundamental rule in differential calculus used to find the derivative of a composite function. A composite function is one function "inside" another, like f(g(x)). The chain rule states that the derivative of 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 dy/dx = dy/du du/dx.

This rule can be visualized as peeling layers off an onion. To find the rate of change of the entire structure (the composite function), you first consider how the outermost layer changes with respect to the layer immediately inside it, and then multiply that by how that inner layer changes with respect to its base (the independent variable). The chain rule is essential for differentiating more complex functions that are not simple power, trigonometric, or exponential functions on their own. It extends the power of differentiation to a vast array of mathematical expressions encountered in physics, engineering, economics, and many other fields where rates of change are analyzed.