Say z^k:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} is a given convex function for each k=1,\dots,K. And, let z:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} be defined as follows: z(x) = \min_{k=1,\dots,K} z^k(x). Intuitively, the directional derivatives of the function $z$ at point $x$ in direction $d$ should match the directional derivatives of the function $z^k$ at point $x$ in direction $d$, where the function $z$ and function $z^k$ coincide at point $x$ (or where the function $z^k$ is active at point $x$). I am seeking a formal proof of this concept in the literature.