  From new integral representations of the $n$-th derivative of Bessel
functions with respect to the order, we derive some reflection formulas for the
first and second order derivative of $J_{\nu }\left( t\right) $ and $% Y_{\nu
}\left( t\right) $ for integral order, and for the $n$-th order derivative of
$I_{\nu }\left( t\right) $ and $K_{\nu }\left( t\right) $ for arbitrary real
order. As an application of the reflection formulas obtained for the first
order derivative, we extend some formulas given in the literature to negative
integral order. Also, as a by-product, we calculate an integral which does not
seem to be reported in the literature.
