I'm trying to compute the derivative of the normalized associated Legendre function of $x=\cos \theta$ and I'm having issues when $\cos \theta =\pm 1$ which causes the denominator to go to $0$. I've seen an answer on the forum about the derivative of the associated Legendre polynomial at the boundary of $x=\pm 1$ but I'm looking for the derivative of the normalized associated Legendre function. The answer is $0$ for all $m$ except for $m=1$ so I was hoping to find a closed form expression in a textbook showing the expression for the special case but I have not found much about the derivative of the normalized associated Legendre function. Thanks.
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