The chain rule is a formula in calculus that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. It is used to find the derivative of a composite function, which is a function that is composed of two or more functions. The rule states that the derivative of the composite function is the inner function within the derivative of the outer function, multiplied by the derivative of the inner function.

The chain rule is expressed mathematically as follows: 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. In Leibniz notation, this is written as dy/dx = dy/du * du/dx.

The chain rule is used to differentiate composite functions, which are functions that can be expressed as a function of a function. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x².

The chain rule is an important tool in calculus, and it is used in many different applications. It is also a fundamental concept in higher-dimensional calculus, where it represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g.