  To target challenges in differentiable optimization we analyze and propose
strategies for derivatives of the Mat\'ern kernel with respect to the
smoothness parameter. This problem is of high interest in Gaussian processes
modelling due to the lack of robust derivatives of the modified Bessel function
of second kind with respect to order. In the current work we focus on newly
identified series expansions for the modified Bessel function of second kind
valid for complex orders. Using these expansions we obtain highly accurate
results using the complex step method. Furthermore, we show that the
evaluations using the recommended expansions are also more efficient than
finite differences.
