$$\begin{align} E &= \frac{1}{2} x^\top D^\top D x + \frac 12 \mu^\top D^\top D \mu - x^\top D^\top \eta \\ \end{align}$$

$$E' = \frac{\partial E}{\partial x} = x^\top D^\top D + \mu^\top D^\top D J_\mu - \eta^\top D - x^\top D^\top J_\eta$$

$$\begin{align} \frac{\partial E'}{\partial t_i} &= (D^\top D)_{:i} + (J_\mu^\top D^\top D J_\mu)_{:i} + \frac{\partial J_\mu^\top}{\partial t_i} D^\top D \mu - (D^\top J_\eta)_{:i} - (J_\eta^\top D^\top)_{:i} - \left \{ x^\top D^\top \frac{\partial J_\eta}{\partial t_i} \right \}^\top \end{align}$$

$$\begin{align} \frac{\partial^2 \mu}{(\partial t_i)^2} &= \frac{\partial}{\partial t_i} (J_\mu)_{:i} \\ &= \frac{\partial}{\partial t_i} \left [\operatorname{diag_{3x1}}(\Delta_{3x3} z) + \left( \Delta_{1x3} \right)^\top \odot \operatorname{repmat}(z_3 , N/3, 1) + K_* J_z \right ]_{:i} \\ &= \frac{\partial}{\partial t_i} \left( \begin{array}{c} 0 \\ 0 \\ \vdots \\ \delta^\top_{i3x3} z \\ \vdots \\ 0 \\ 0 \end{array} \right ) + \delta_{i,3x1} \odot \operatorname{repmat}(z^{(3)}_i, N/3, 1) + K_* (J_z )_{:i} \\ &= \left( \begin{array}{c} 0 \\ 0 \\ \vdots \\ C^\top_{i3x3} z \\ \vdots \\ 0 \\ 0 \end{array} \right ) + \left ( \begin{array}{c} 0 \\ 0 \\ \vdots \\ \delta^\top_{i3x3} z'_i \\ \vdots \\ 0 \\ 0 \end{array} \right ) + C_{i,3x1} \odot \operatorname{repmat}(z^{(3)}_i, N/3, 1) + \delta_{i,3x1} \odot \operatorname{repmat}(z'^{(3)}_i, N/3, 1) + K_* (J'_z )_{:i} \\ + K'_* (J_z )_{:i} \\ \end{align}$$