In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. If a function f maps a variable u to f(u), and another function g maps a variable x to g(x), then the composition of f with g is f(g(x)). The chain rule states that the derivative of this composite function is the derivative of f with respect to u multiplied by the derivative of g with respect to x. This can be written as: d/dx(f(g(x))) = f'(g(x)) * g'(x).
The chain rule was first used by Gottfried Wilhelm Leibniz in 1676, who is also credited with the invention of calculus along with Isaac Newton. The notation d/dx is known as Leibniz's notation.
The chain rule is a fundamental concept in calculus and is used extensively in physics, engineering, and other fields that use calculus to model the world. In machine learning, the chain rule is the mathematical foundation of backpropagation, the algorithm used to train artificial neural networks.