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\begin{equation}  k(t)=\left(\frac{dS(t)^{-1}}{dt}\right)  \end{equation}  The same can be derived for $n^{th}$ order reactions, where $A+nA\rightarrow B$ B$. The integrated rate law of these reactions is  \begin{equation}  \frac{1}{[A_t]^{n-1}}=\frac{1}{[A_0]^{n-1}}+(n-1)\omega t  \end{equation}  From the integrated rate law, we obtain the survival function  \begin{equation}  S(t)=\frac{[A_t]}{[A_0]}\sqrt[n-1]{\frac{1}{1+(n-1)\omega t[A_0]^{n-1}}}