Jim Gatheral’s book “The Volatility Surface – A Pratitioner’s Guide” (Wiley, 2006) provides an excellent treatment on volatility modelling. Most of the proofs and derivations are only outlined in the book and it is left to the reader to do the intermediate steps. The attached document provides the missing steps in the derivation of the local volatility as a function of the implied volatility on pages 11 – 13 in the book.
A full derivation of the Dupire equation along the lines of Gatheral’s book can for example be found on Fabrice Rouah’s webpage.
Gatheral, Jim (2004) The Volatility Surface: A Practitioner’s Guide: Wiley Finance