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.
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.