#Contents# * table of contents {:toc} ## Definition ## 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 $$\Del_{w}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x w) - f(v + y w)}{x - y}$$ ## See also ## * [[sequentially Cauchy complete Archimedean ordered field]] * [[real vector space]] * [[Newton-Leibniz operator]] * [[partial derivative]]