Find the derivative of
So I'm guessing here you just use the chain rule to find the derivative. I get:
If I know I did this much correct so far I can go on and simplify it.
The derivative of is quoted in all the reference works and textbooks that I've ever come across as . IMO it's perfectly appropriate, and I actually find it easier to learn like that. As you say, it saves a quotient.
To take it to its logical conclusion, there's no point in learning it anyway, because you can always use the quotient rule on .