\begin{align}
     \frac{\partial}{\partial \xi^{k,t}_j} \log p(W_{\mathcal{D}(1:G,t)} | \phi^{\cdot,t}, \psi^{\cdot,t}) 
     &= \sum_{v=1}^V \rho_{v,j} \frac{\partial}{\partial  \psi^{k,t}_v} \log p(W_{\mathcal{D}(1:G,t)} | \phi^{\cdot,t}, \psi^{\cdot,t}) \nonumber \\
     &= \sum_{d \in \mathcal{D}(1:G,t)} \frac {\tilde{\phi}_k^{\gamma_d,t} \prod_{i=i_1}^{i_{L_d}} \tilde{\psi}_{w_{d,i}}^{k,t}} {\sum_{l=1}^K \tilde{\phi}_l^{\gamma_d,t} \prod_{i=i_1}^{i_{L_d}} \tilde{\psi}_{w_{d,i}}^{l,t}} \left( \sum_{i=i_1}^{i_{L_d}} \sum_{v=1}^V \rho_{v,j} \mathbb{I}(v=w_{d,i}) - L_d \sum_{v=1}^V \rho_{v,j} \tilde{\psi}_{v}^{k,t} \right) \nonumber \\
     &= \sum_{d \in \mathcal{D}(1:G,t)} \frac {\tilde{\phi}_k^{\gamma_d,t} \prod_{i=i_1}^{i_{L_d}} \tilde{\psi}_{w_{d,i}}^{k,t}} {\sum_{l=1}^K \tilde{\phi}_l^{\gamma_d,t} \prod_{i=i_1}^{i_{L_d}} \tilde{\psi}_{w_{d,i}}^{l,t}} \left( \sum_{i=i_1}^{i_{L_d}} \rho_{w_{d,i},j} - L_d \rho_{\cdot,j}^\text{T} \tilde{\psi}^{k,t} \right) \label{eq:diff_log_p_xi_final}
\end{align}
