Let $f_n$ ($n=1,2,\dots$) be a sequence of functions $f_n\colon \mathbb R\to \mathbb R$ of class $C^1$ such that $f_n \rightrightarrows 0 $, $f_n' \rightrightarrows 0 $. Assume moreover that functions $f_n(\sqrt{x})$ ($n=1,2,...$) are also of class $C^1[0, \infty)$.
Is it then $[f_n(\sqrt{x})]' \rightrightarrows 0$ ?