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k(t)=\left(\frac{dS(t)^{-1}}{dt}\right)  \end{equation}  The same can be derived for $n^{th}$ order reactions, where $A+nA\rightarrow B$. 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}