Let \(p(t)\) be the probability of survival from 0 to \(t\). The probability of surviving to \(s\) given survival to \(t\) is then

\[p(s \mid t)=\frac{p(s)}{p(t)}\]

Let \[f(t)=-\frac{p^{\prime}(t)}{p(t)}\]
Show that \[p(s \mid t)=e^{\int_{t}^{s} f(u) \mathrm{d} u}\]