The Chain Rule
Point of Post: In this post we discuss the chain rule of total derivatives which generalizes the normal chain rule.
If the total derivative is the generalization of the normal derivative for functions we’ve made it out to be one would hope that it shares most of the nice attributes of the regular derivative. In particular, one of the nicest properties of the normal derivatives for real valued real functions is the chain rule. Here we prove that an analogous theorem holds for differentiable mappings .
Our goal is to show that if and are maps and they share a ‘common point’ of differentiability then their composition is differentiable and derive a formula for it. Indeed:
Theorem (Chain Rule): Let and be maps with , and suppose that is differentiable at and is differentiable at , then is differentiable at and
Or, in Jacobian notation
We know then that
Moreover, we see that
But, since if and only if
it suffices to show that
The second of these follows immediately from the fact that
where is the operator norm. To show the other limit is zero we note that for any there exists some such that if . But, we may choose such that implies (since is continuous at ). Thus, we get that for that
But, dividing both sides by and letting finishes the argument (really what we have to do formally is divide both sides by and choose a revised restriction on such that the right-hand expression is less than , but this is standard since ).
1. Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.