  One shortcoming of the chain rule is that it does not iterate: it gives the
derivative of f(g(x)), but not (directly) the second or higher-order
derivatives. We present iterated differentials and a version of the
multivariable chain rule which iterates to any desired level of derivative. We
first present this material informally, and later discuss how to make it
rigorous (a discussion which touches on formal foundations of calculus). We
also suggest a finite calculus chain rule (contrary to Graham, Knuth and
Patashnik's claim that "there's no corresponding chain rule of finite
calculus").
