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. The rule is used to find the derivative of a composite function, which is a function that is constructed by applying one function to the output of another function. The chain rule states that if g is a function that is differentiable at a point c and f is a function that is differentiable at g(c), then the composite function is differentiable at c, and the derivative is given by the product of the derivative of f with respect to its input and the derivative of g with respect to its input.

The chain rule can be applied to composites of more than two functions. To take the derivative of a composite of more than two functions, the chain rule states that it is sufficient to compute the derivative of the outermost function and the derivative of the innermost function, and then apply the chain rule repeatedly to the intermediate functions.

The chain rule can be written in different forms, including the total derivative, which is a linear transformation that captures all directional derivatives in a single formula. The total derivative can be used to find the derivative of a composite function of differentiable functions f : Rm → Rk and g : Rn → Rm, and a point a in Rn.

In summary, the chain rule is a formula that allows us to find the derivative of a composite function by taking the derivative of the outermost function and the derivative of the innermost function, and then multiplying them together. The rule can be applied to composites of more than two functions, and can be written in different forms, including the total derivative.