Around the formula of Faà di Bruno

The reader will find in this book both elementary and non-elementary mathematics. We tried to limit the volume of non-elementary facts, in order to increase the audience of the present work. Our main goal is to present in an extended framework the famous formula of Faà di Bruno which deals with higher order derivatives of composite functions. To this end, we begin with a rehearsal of some elementary and non-elementary facts concerning real functions of a real variable. Then we present and prove in a detailed manner the formula of Faà di Bruno. The proof in this book is original (we recognize the fact that it is a bit too long, but we hope it is clear and self-contained). Then we use the formula to obtain expressions for the higher derivatives of inverse functions and of functions which depend implicitly on other functions. Parametric representations appear as natural applications and are presented in detail with examples.