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If $X$ has a continuous cumulative distribution function $F$, then $F(X)$ has the uniform distribution on $(0,1)$. Combined with inverse transform sampling, this constructs the monotone rearrangement $G^{-1}(F(X))$ to a target with quantile function $G^{-1}$.

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.