Inspired by The Theoretical Minimum: What You Need to Know to Start Doing Physics: here’s a better proof/justification for the product rule in differential calculus than the one I set out here last month.
We will start with what we will treat as an axiomatic definition of the differential of the function :
In this case we have , so
From our definition we can substitute for and and simplifying our notation for presentational reasons so that etc:
Giving (after dividing through by ):
As the first term falls to zero and so we are left with:
Which, of course, is the product rule.
Update: See this most excellent comment from Professor Rubin.