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Derive the general equation for failure-time distribution
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To derive the relationship expressing the failure-time density in terms of the failure-rate function. Make use of the fact that $R(t)=1-F(t)$ and, hence, that $F^{\prime}(t)=-R^{\prime}(t)$, we can write
$Z(t)=-\frac{R^{\prime}(t)}{R(t)}=-\frac{d[\ln R(t)]}{d t}$
Solving this differential equation for $R(t)$, we obtain
$R(t)=e^{-\int_{0}^{t}} Z(x) d x$
and, making use of the relation $f(t)=Z(t) \cdot R(t)$, we finally get

$f(t)=Z(t) \cdot e^{-\int_{0}^{t} Z(x) d x}$
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