Derivation/Extension to localization/Exercise

Let ${}R$ denote a commutative ${}K$-algebra and let ${}W\subseteq R$ denote a multiplicative system. Let ${}D\colon R\rightarrow R$ denote a ${}K$-derivation. Show that we get via

${}D{\left({\frac {f}{g}}\right)}:={\frac {gD(f)-fD(g)}{g^{2}}}\,$

a derivation on the localization ${}R_{W}$ which extends ${}D$.

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