We extend a Rockefellers result for the subdifferential of the upper envelope $j=\sup\limits_{1\leq i\leq n}f$, of a finite collection $f_1,\dots,f_r$ of convex proper functionals on a locally convex Hausdorff topological space $X$. Assuming that $f_1,\dots,f_{n-1}$ are finite and continuous at a point $x_0$ of $X$ where $f_n$ is finite, we show that, for any point $x$ of $X$ such that $f(x)$ is finite \begin{equation}\partial f(x)=\mathrm{co}\{\partial f_k(x)\,:\, f_k(x)=f(x)\}+\sum\limits_{i=1}^n N(\mathrm{dom\ }f_i, x),\tag{*}\end{equation} where co stands for the convex hull and $N(\mathrm{dom\ }f_i, x)$ for the normal cone to the domain dom $f_i$ of $f_i$ at $x$. We also give an application of (*) to asymptotical analysis related to a result by Choquet, and prove that (*) remains true when the epigraph of the Legendre-Fenchel conjugate of $f$ is weak* complete and pointed, and the $f_i$ are lower-semicontinuous.