= Solution
The <cumulative distribution function> is $F(x)=P(X\leq x)$. For the differentiable distribution in the question, its <probability density function> is $f(x)=F'(x)$, so <probabilities> over intervals are obtained by integrating $f$. The survival <probability> is $P(X>x)=1-F(x)$. Differentiating immediately gives
$$
\boxed{f(x)=-\frac d{dx}P(X>x)}.
$$
The strict inequality is consistent with the definition using $X\leq x$. This relation connects the <probability density function> with the decline of the survival <probability>.
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