When reading a paper, I found a theorem stated as
Theorem 1 (Radmacher, [22, Theorem 3.1.6]) If $f:\mathbb R^n\to\mathbb R^m$ is locally Lipschitz continuous function, then $f$ is differentiable a.e. Moreover, if $f$ is Lipschitz continuous, then $$L(f)=\sup_{x\in\mathbb R^n}||D_xf||_2$$ where $||M||_2=\sup_{\{x:||x|=1\}}||Mx||_2$ is the operator norm of the matrix $M\in\mathbb R^{m\times n}$.
I can find various proofs for the first statement. But it is hard to find the proof of the second statement (that the optimal Lipschitz constant is the supremum of the Jacoian matrix $D_xf$). The original reference book by Herbert Federer doesn't explain the second statement explicitly.
Is there any reference to the proof of the second statement? Or is there anyone to prove the second statement?