Chain Rule

The chain rule gives the derivative of a function composition. If is differentiable at the point and is differentiable at the point , then is differentiable at . Furthermore, let and , then

(1)

There are a number of related results that also go under the name of "chain rules." For example, for differentiable functions , , and , if , , and , then

(2)

The "general" chain rule applies to two sets of differentiable functions

			(3)
			(4)
			(5)

and

			(6)
			(7)
			(8)

Defining the Jacobian matrix by

(9)

and similarly for and , then gives

(10)

In differential form, this becomes

(11)

(Kaplan 1984).

The term "chain rule" is also used for the probability chain rule, which factors a joint probability into conditional probabilities, and for the entropy chain rule in information theory. These are distinct from the chain rule for derivatives.