The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, ¶P_n^m(z)/¶m{\partial {P_{n}^{\mu}(z)}/\partial\mu}, is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of [¶P_n^m(z)/¶m]_{m = ±m}{[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=\pm m}}, where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, Q_n^±m(z){Q_{n}^{\pm m}(z)}. In particular, we arrive at formulas which generalize to the case of Q_n^±m(z){Q_{n}^{\pm m}(z)} (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives [¶² P_n^m(z)/¶m²]_{m = m},[¶Q_n^m(z)/¶m]_{m = m}{{[\partial^{2} P_{n}^{\mu}(z)/\partial\mu^{2}]_{\mu=m}},{[\partial Q_{n}^{\mu}(z)/\partial\mu]_{\mu=m}}} and [¶Q_-n-1^m(z)/¶m]_{m = m}{[\partial Q_{-n-1}^{\mu}(z)/\partial\mu]_{\mu=m}}, all with m > n, are also evaluated.