The authors discuss the double zeros of the derivatives of cylindrical functions, say \(J'_{\nu}\), \(Y'_{\nu}\), and \(H^{(1)}_{\nu}\). First they summarize the fundamental properties. \(J'_{\nu}(z)\) has no double zeros even in the complex variables of \(\nu\) and z, provided that \(| \arg \nu | \leq \pi /2\) (Theorem 1). For each positive integer n, there is a unique \(\nu_ n\) in the interval (-n-1,-n) for which \(J'_{\nu_ n}\) has multiple zeros, and \(\nu_ n=-n-1/6+o(1),\) \(n\to \infty\). (Theorem 2). \(Y'_{\nu}(z)\) in \(| \arg z| \leq \pi\) has double zeros only for particular values \(\nu_ n=\alpha_ n+i\beta_ n\) with the asymptotic behavior \[ \alpha_ n=n+1/3+p\quad n^{- 2/3}+o(n^{-2/3}),\quad \beta_ n=a-\sqrt{3}p\quad n^{-2/3}+o(n^{- 2/3}+o(n^{-2/3}), \] \(\beta_ n=a-\sqrt{3}pn^{-2/3}+o(n^{2/3}),\quad n\to \infty,\) where \(p=\sqrt{3}\Gamma (1/3)/20\pi 6^{1/3}\Gamma (2/3)=- 0.030012575564\) and \(a=\ln 2/2=0.110317800076\); \(\nu_ 0\) is close to \(1/3+2ia\). (Theorem 3). \(H_{\nu}^{(1)'}\) has double zeros only for \(\nu_ n=\gamma_ n+i\xi_ n\), \(\gamma_ n>0\), \(\xi_ n>0\) with \[ \gamma_ n=n+1/3+p\cdot n^{-2/3}+o(n^{-2/3}),\quad \xi_ n=- \sqrt{3}p\quad n^{-2/3}+o(n^{-2/3}),\quad n\to \infty. \] \(\nu_ 0\) is close to \(1/3+i5\sqrt{3}| p| /2\) (Theorem 4). They give numerical tables of \(\alpha_ n+i\beta_ n\) and \(\gamma_ n+i\xi_ n\) both for \(0\leq n\leq 100\), which seems an important contribution. The computations are mainly due to L. M. Delves and J. N. Lyness’ method [Math. Comput. 21, 543-560 (1967; Zbl 0153.179)] to determine the zeros of an analytic function by virtue of complex integral, starting from the above asymptotic values.