Quadratic-mean to L1 density derivative
= Quadratic-mean to L1 density derivative
{title2=$\|p_t-p-tpg\|_1=o(|t|)$}
Let $\delta_t=\sqrt{p_t}-\sqrt p=(t/2)g\sqrt p+r_t$, with $\|r_t\|_2=o(|t|)$. Since $p_t-p=2\sqrt p\,\delta_t+\delta_t^2$, the <Cauchy-Schwarz inequality> gives $\|p_t-p-tpg\|_1\leq2\|r_t\|_2+\|\delta_t\|_2^2=o(|t|)$. Thus <differentiability in quadratic mean> suffices to differentiate bounded density <integrals>, even when the density itself has no useful pointwise <derivative>.