Let $U \subset \mathbb{R}^n$ be open, let $f: U \to \mathbb{R}$ be a function, let $a \in U$, and let $v \in \mathbb{R}^n$. Let $I \subset \mathbb{R}$ be an open interval with $0 \in I$ such that $a + tv \in U$ for every $t \in I$, and define the line restriction
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\begin{align*}
g: I \to \mathbb{R}, \quad t \mapsto f(a + tv).
\end{align*}
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Then the [directional derivative](/page/Directional%20Derivative) $D_v f(a)$ exists if and only if the ordinary derivative $g'(0)$ exists. In that case,