  We classify the derivations of degree-one generalized Weyl algebras over a
univariate Laurent polynomial ring. In particular, our results cover the
Weyl-Hayashi algebra, a quantization of the first Weyl algebra arising as a
primitive factor algebra of $U_q^+(\mathfrak{so}_5)$, and a family of algebras
which localize to the group algebra of the infinite group with generators $x$
and $y$, subject to the relation $xy = y^{-1}x.$
