Let $f : \mathbb{R}^{N \times 1} \to \mathbb{R}$ be a D5614: Differentiable real function at $x_0 \in \mathbb{R}^{N \times 1}$.
Then
\begin{equation}
\Vert \nabla f(x_0) \Vert_2
= \underset{u \in \mathbb{R}^{N \times 1} : \Vert u \Vert_2 = 1}{\text{max}} \, D_u f(x_0)
\end{equation}