Ableitungssatz

The Ableitungssatz, also known as the Chain Rule, is a fundamental theorem in calculus that describes the derivative of a composite function. It states that if a function f is composed of two functions u and v, such that f(x) = u(v(x)), then the derivative of f with respect to x is given by the product of the derivative of u with respect to v and the derivative of v with respect to x. Mathematically, this is expressed as:

f'(x) = u'(v(x)) * v'(x)

The Ableitungssatz is particularly useful in situations where a function is defined in terms of other functions, and it allows for the computation of the derivative of the composite function without needing to expand or simplify the function explicitly. It is a cornerstone of calculus and is widely used in various fields, including physics, engineering, and economics, to analyze the behavior of functions and their derivatives. The theorem can be extended to composite functions involving more than two functions, and its proof relies on the definition of the derivative and the properties of limits.