  We prove that for $\nu>n-1$ all zeros of the $n$th derivative of Bessel
function of the first kind $J_{\nu}$ are real and simple. Moreover, we show
that the positive zeros of the $n$th and $(n+1)$th derivative of Bessel
function of the first kind $J_{\nu}$ are interlacing when $\nu\geq n,$ and $n$
is a natural number or zero. Our methods include the Weierstrassian
representation of the $n$th derivative, properties of the Laguerre-P\'olya
class of entire functions, and the Laguerre inequalities. Some similar results
for the zeros of the first and second derivative of the Struve function of the
first kind $\mathbf{H}_{\nu}$ are also proved. The main results obtained in
this paper generalize and complement some classical results on the zeros of
Bessel functions of the first kind. Some open problems related to Hurwitz
theorem on the zeros of Bessel functions are also proposed, which may be of
interest for further research.
