  We calculate the derivative of the $\mathrm{ber}_{\nu }$,
$\,\mathrm{bei}_{\nu }$, $\mathrm{ker}_{\nu }$, and $\,\mathrm{kei}_{\nu }$
functions with respect to the order $\nu $ in closed-form for $\nu \in
\mathbb{R}$. Unlike the expressions found in the literature for order
derivatives of the $\mathrm{ber}_{\nu }$ and $\,\mathrm{bei}_{\nu }$ functions,
we provide much more simple expressions that are also applicable for negative
integral order. The expressions for the order derivatives of the
$\mathrm{ker}_{\nu }$ and $\,\mathrm{kei}_{\nu }$ functions seem to be novel.
Also, as a by-product, we calculate some new integrals involving the
$\mathrm{ber}_{\nu }$ and $\,\mathrm{bei}_{\nu }$ functions in closed-form.
Finally, we include a simple derivation of some integral representations of the
$\mathrm{ber}$ and $\,\mathrm{bei}$ functions.
