I am reading The Matrix Cookbook, which has the following for the Frobenius norm
$$ \frac{\partial }{\partial X} \| X\|_{F}^{2} = \frac{\partial}{\partial X}\textrm{Tr}(XX^H) = 2 X$$
If, in general, the norm $\| \cdot \|_{p,q}$ is given by
$$ \| X \|_{p,q} = \Bigg( \sum_{j=1}^{n} \Bigg( \sum_{i=1}^{m} |x_{ij}|^{p} \Bigg)^{\frac{q}{p}} \Bigg)^{\frac{1}{q}} $$
how would you go about finding $\frac{\partial}{\partial X}\| X\|_{p,q} $?