Question: For a continuously differentiable survival function (S(t)), the hazard function is defined as (h(t)=-frac{S^{prime}(t)}{S(t)}). Prove that (S(t)=exp left(-int_{0}^{t} h(s) d s ight)).

For a continuously differentiable survival function $S(t)$, the hazard function is defined as $h(t)=-\frac{S^{\prime}(t)}{S(t)}$. Prove that $S(t)=\exp \left(-\int_{0}^{t} h(s) d s\right)$.

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