Let $m \in \mathbb{N}$, let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}$ be differentiable at $a$ with respect to the Euclidean norm $|\cdot|$ on $\mathbb{R}^m$. Let
denote the total derivative of $f$ at $a$, and let $\cdot$ denote the Euclidean dot product on $\mathbb{R}^m$. Then the gradient $\nabla f(a)$ exists and, for every $h \in \mathbb{R}^m$,
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\begin{align*}
Df_a(h)=\nabla f(a)\cdot h.
\end{align*}