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In order to determine $k(t)$ in higher order reactions, we use the survival function, but it is not necessary. From the survival function, the time dependent rate coefficient is  \begin{equation}  k_i(t) \equiv \frac{d}{dt}\frac{1}{(S(t)^{i-1)}} \frac{d}{dt}\frac{1}{S(t)^{i-1}}  \end{equation}  Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to to second order irreversible decay. The integrated rate law in second order is  \begin{equation}