Let $f : \mathbb{R}^{N \times 1} \to \mathbb{R}$ be a D5614: Differentiable real function at $x_0 \in \mathbb{R}^{N \times 1}$.
Let $u \in \mathbb{R}^{N \times 1}$ be a D5200: Real column matrix such that

(i)	\begin{equation} \Vert u \Vert_2 = 1 \end{equation}

Then

(1)	\begin{equation} D_u f(x_0) \leq \Vert \nabla f(x_0) \Vert_2 \end{equation}
(2)	\begin{equation} D_u f(x_0) = \Vert \nabla f(x_0) \Vert_2 \quad \iff \quad \Vert \nabla f(x_0) \Vert_2 u = \nabla f(x_0) \end{equation}