Derivative of Associated Legendre polynomials at $x = \pm 1$
The singularity at the denominator can be eliminated using L'Hospital's theorem, once you notice that the associated Legendre function has value of $0$ at $\pm 1$.
Maybe this is not a right solution, because I found another formula about the derivative of the associated Legendre function here,
- Spherical Harmonics
and it gives a difference solution when I apply the same method.