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For integral representations of associated Legendre functions in terms ofmodified Bessel functions, we establish justification for differentiation underthe integral sign with respect to parameters. With this justification,derivatives for associated Legendre functions of the first and second kind withrespect to the degree are evaluated at odd-half-integer degrees, for generalcomplex-orders, and derivatives with respect to the order are evaluated atinteger-orders, for general complex-degrees. We also discuss the properties ofthe complex function $f:{\mathbb C}\setminus\{-1,1\}\to{\mathbb C}$ given by$f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.

On Parameter Differentiation for Integral Representations of Associated Legendre Functions