I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes
http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf
Suppose, $R$ is a commutative ring, $W$ a multiplicatively closed subset in $R$, $M$ an $R$-module. If $D:R\to M$ is a derivation, then $W^{-1}D: W^{-1}R\to W^{-1}M$ is a derivation where $W^{-1}D$ acts on $\frac{r}{w}$ by the quotient rule, i.e. maps $\frac{r}{w}$ to $\frac{wD(r)-rD(w)}{w^2}$.
I have tried several manipulations, but I am unable to show that this map is well defined. I would appreciate if anyone can help me see what I am missing here.
Thanks.