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The time-dependent rate coefficient, $k(t)$, is determined by integrating the rate law of the reaction and forming a survival function from the integrated rate law.  In order to determine $k(t)$ in higher order reactions, we use the survival function, but it is not necessary. The survival function of any reaction involving any number of the same molecule can be derived using the integrated rate laws of reactions. For example, the $n^{th}$ order integrated rate law is  \begin{equation}  \frac{1}{[c_A(t)]^{n-1}} = \frac{1}{[c_A(0)]^{n-1}}+(n-1)\omega t  \end{equation}  From the integrated rate law, we get the survival function  \begin{equation}  S(t)=\frac{C_A(t)}{C_A(0)}=\sqrt[n-1]{\frac{1}{1+(n-1)\omega tC_A(0)^{n-1}}}  \end{equation}  From the survival function, the time dependent rate coefficient is  \begin{equation}  k(t) = (\frac{d\frac{1}{(S(t)^{n-1)}}}{dt})  \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}  C_A(t) = \frac{C_A(0)}{1+\omega tC_A(0)}  \end{equation}  The second order survival function is  \begin{equation}  S(t) = \frac{C_A(t)}{C_A(0)}=\frac{1}{1+\omega tC_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)