Series reversion as the reversed chain rule

I recently had occasion to extend a derivative-computation package I had written in 1971 to add the capability of dealing with a function defined implicitly. This led me to see the duality between Taylor series reversion and the chain rule of differentiation of a composite function. I was also struck with the compactness with which these algorithms could be expressed in a programming language, which in my case was Fortran 77. The basic ideas involved will probably not be new to persons who have worked on computerization of symbolic mathematics or other approaches to derivative computation, however, I think this may be of interest to many readers of this newsletter as an instance of a very small body of code implementing a mathematical transformation that might a priori be thought to be somewhat complicated.