## 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 which 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)

and
 (4)

Defining the Jacobi Matrix by
 (5)

and similarly for and then gives
 (6)

In differential form, this becomes

 (7)

(Kaplan 1984).

Kaplan, W. Derivatives and Differentials of Composite Functions'' and The General Chain Rule.'' §2.8 and 2.9 in Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 101-105 and 106-110, 1984.