If $ \phi, \varphi \in \mathcal{D} $ are given test functions and $ \text{Supp} \ \varphi \subseteq \text{Supp} \ \phi $. I have used many times in my PDE class that the quotient $ \varphi / \phi $ is well-defined and smooth because $ \phi $ vanishes on sets where $ \varphi $ vanishes.
I have never quiet thought this through. What is the right point of view in looking at the well-definedness of this function? Is it that we assign the value zero to the expression $0/0$, or perhaps any other reason? Is this statement even correct, in the sense that there has to be some additional assumption on the support? In particular, if $ \phi \in \mathcal{D}$. Then since $ \text{Supp} \ \phi' \subseteq \text{Supp} \ \phi $ , does it make sense to write $ \phi' / \phi$?
Many thanks in advance!