The spherical harmonics, $Y^m_n(\theta, \varphi)$, are only defined when $m$ $\in$ $[n,n-1,...,-n+1,-n]$. However, the derivative relation with respect to $\theta$ requires $Y^{m+1}_n(\theta, \varphi)$ (see http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/ShowAll.html).
Does this mean that $\frac{\partial Y^m_n(\theta, \varphi)}{\partial \theta}$ is not defined for $n = m$?
I may have some misunderstanding here.