In this Wolfram Blog entry, they give this ridiculously complicated expression for the first derivative of the Bessel function $J_n(x)$ with respect to $n$ that uses higher hypergeometric functions.
I can't believe that a derivative can be so complicated, but by searching I could not find any results for the derivative. Differentiation should be easy and mechanical. This looks more complicated than integration.
Is there really no simpler form?
Recall that Bessel functions can be represented in terms of the ${}_0 F_1$ hypergeometric function. In general, differentiation of hypergeometric functions ${}_p F_q$ with respect to their numerator or denominator parameters requires the use of more complicated functions like the multivariate Kampé de Fériet function, or else do not yet have known closed forms. From the series viewpoint, your differentiation turns the Pochhammer symbols into polygamma functions/generalized harmonic numbers, and sums with those factors are often difficult to sum in closed form.
As a further example of the difficulties I described, see this article (derivatives of ${}_2 F_1$), or this article (derivatives of ${}_1 F_1$). You'd do well not to underestimate the complications of dealing with special functions, more so deriving expressions that are sensible for as wide a parameter range as possible.