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.
OK, back to the issue of Euler’s number and the proof that it is . This is a proof based on, though expanded from, what I picked up from a rather excellent website here.
Once again we start from the proposition that there exists a number such that and hence (see here for why the second follows from the first).
Here we go…
At this point we still have , an indeterminate number, so we need to look for a determinate form.
So we take this as:
Now we have , another indeterminate, but we can also apply L’Hospital’s rule.
This states that, (a proof to follow sometime).
Here , and, of course, .
So, using the chain rule, , for the numerator, , : so
And for the denominator , giving for the whole thing,
Hence , so .
This is just an online note to myself about differential calculus. A level maths again…
Using the chain rule:
And , Euler’s number, really is magical.