Chain Rule

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, if , , and , then

(2)

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

			(3)
			(4)
			(5)

and

			(6)
			(7)
			(8)

Defining the Jacobi rotation matrix by

(9)

and similarly for and , then gives

(10)

In differential form, this becomes

(11)

(Kaplan 1984).