Date of this Version
A "from scratch" proof of the differentiability of ax, a > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define ax as ex log a its differentiability and other properties following from that of the functions ex and log x. Unfortunately, the usual definitions of ex and log x involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of e, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is told that e is e1. He would have a much better feeling for the "naturalness" of e if it were defined as that number a for which (ax)' = ax.
The purpose of this note is to provide a direct and relatively simple way of getting at the differentiability of ax.