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Given a sequentially Cauchy complete Archimedean ordered field $\mathbb{R}$ of scalars, let $V$ be a $\mathbb{R}$-real vector space $V$, and let $f:V \to \mathbb{R}$ be a directionally differentiable function valued in scalars. Given a vector $w:V$, the **directional derivative** $\Del_{w}$ is pointwise defined as

$\Del_{w}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x w) - f(v + y w)}{x - y}$