Quantile derivative identity
= Quantile derivative identity
{title2=$q^{\prime}(u)=1/f(q(u))$}
If $q$ is the <quantile function> of an absolutely continuous <probability distribution> with <probability density function> $f$, then $q^{\prime}(u)=1/f(q(u))$ for almost every $u\in(0,1)$. Gaps in the support may give jumps in $q$, so the identity concerns its ordinary almost-everywhere derivative. It follows by differentiating the inverse relation to the <cumulative distribution function>.