$$\prod_i \frac{\prod_j\exp(a_{ij}\theta- \delta_{ij})}{\sum_h\exp(a_{ih}\theta- \delta_{ih})}$$
Taking log
$$\sum_i\sum_j {(a_{ij}\theta- \delta_{ij})}-{\sum_i\log\sum_h\exp(a_{ih}\theta- \delta_{ih})}$$
Derivative with respect to $\theta$
$$\sum_i\sum_j {(a_{ij})}-{\sum_i\frac{(a_{ih}\exp((a_{ih}\theta- \delta_{ih}))}{\sum_h\exp(a_{ih}\theta- \delta_{ih})}}$$