# Separable function

## Details

Multiplicatively separable

(1)$F(\Theta,\mathbf{x})=\prod_{i=1:n} f_i(\theta_i,x_i)$

Then

(2)$\frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\frac{\partial f_j(\theta_j,x_j)}{\partial \theta_j} \prod_{i=1:n,i \ne j} f_i(\theta_i,x_i)$
(3)$\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\frac{\partial^2 f_j(\theta_j,x_j)}{\partial \theta_j^2} \prod_{i=1:n,i \ne j} f_i(\theta_i,x_i)$
(4)$\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\frac{\partial f_j(\theta_j,x_j)}{\partial \theta_j} \frac{\partial f_k(\theta_k,x_k)}{\partial \theta_k} \prod_{i=1:n,i \ne j,k} f_i(\theta_i,x_i)$

Special case:

(5)$F(\Theta,\mathbf{x})=\exp \sum_{i=1:n} f_i(\theta_i,x_i)$

Then

(6)$\frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\exp (\frac{\partial f_j(\theta_j,x_j)}{\partial \theta_j} +\sum_{i=1:n,i \ne j} f_i(\theta_i,x_i))$
(7)$\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\exp (\frac{\partial^2 f_j(\theta_j,x_j)}{\partial \theta_j^2} +\sum_{i=1:n,i \ne j} f_i(\theta_i,x_i) )$
(8)$\frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\exp (\frac{\partial f_j(\theta_j,x_j)}{\partial \theta_j} \frac{\partial f_k(\theta_k,x_k)}{\partial \theta_k} +\sum_{i=1:n,i \ne j,k} f_i(\theta_i,x_i) )$