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\begin{equation}  S(t)=\frac{[A_t]}{[A_0]}=\frac{1}{1+\omega t[A_0]}  \end{equation}  From this survival function, taking the inverse of the survival function and then the time derivative yields the time dependent rate coefficient.  \begin{equation}  k(t)=\left(\frac{dS(t)^{-1}}{dt}\right)  \end{equation}