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\begin{equation}  S(t) = \frac{C_A(t)}{C_A(0)}.  \end{equation}  From the survival function, the time dependent rate coefficient is determined by taking various time derivatives of the survival function, depending on the total order of reaction. For first order irreversible decay reactions, $A\to B$, the rate law defines  the time dependent rate coefficientis[insert citation]  \begin{equation}  \frac{-d\ln S(t)}{dt} = k(t)  \end{equation}  In traditional kinetics, irreversible decay is only dependent on one rate coefficient, $\omega$. The rate law of a first-order reaction in which $A$ irreversibly decays to B is   \begin{equation}  \frac{-dS(t)}{dt} = \omega S(t)  \end{equation} $k(t)\to\omega$.  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