Note that from $k \, dx = -dx / (1 + dx)$ you *cannot* conclude $k = -1/(1 + dx)$ because the law of cancelation does not apply. Recall the law of cancelation: “if $a, b \in R$ such that $a \, dx = b \, dx$ for all $dx \in \Delta$ then $a = b$.” In our case we have a problem: we can take $a = k$, but $b$ cannot be $-1/(1 + dx)$ because it is supposed to be a fixed real, independent of $dx$.

Does this dispell the mystery?

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