As we are analyzing survival rates, a geometric mean is better used to describe average effects.
By weighting the geometric mean with constellation sizes, we get:
\begin{align}
    R_G = \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
\end{align}
The marginal effect is assumed to be negative, thus-
\begin{align}
    0 > \parder{R_G}{s^i_t}{} =& \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
     \left[  \parder{}{s^i_t}{} \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right] \\
    0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
                                \left( \ln(1-l^i)-
                                        - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
                                        - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
                                \right)-
                                - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
    0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
                                \left( \ln(R_i)-
                                        - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
                                        - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
                                \right)-
                                - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
    0 > &  \ln R_i  - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \sum^N_{j=1} \frac{s_t^j}{S_t} \ln(R_j)  \\
    0 > &  \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G  \\
    \ln \frac{R_G}{R_i} =& \ln R_G - \ln R_i  > - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}--
\end{align}
