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 :

as

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.

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Politicians (“tax revenues [cough][cough] equal [cough][cough] public expenditures”) and IT managers (“project launch date [cough][cough] equals [cough][cough] target date”) share a definition of “equals” different from the mathematical one. Unfortunately, that also applies to the line , where “=” only means “approximates for small “. Using “little-oh” notation (https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation), you can clean this up a bit. , and similarly for . All the terms (and products of them) can be gathered into a single composite term which, after division by , dies at the limit.

Thanks. I’ve updated the OP with a link to your comment.