#Contents# * table of contents {:toc} ## Definition ## Given a [[sequentially Cauchy complete Archimedean ordered field]] $\mathbb{R}$ of scalars and a type of indices $I$, one could define a [[real vector space]] $V \coloneqq \mathbb{R}^I$ with a basis vector function $e:I \hookrightarrow V$. Let $f:V \to \mathbb{R}$ be a [[differentiable function|differentiable]] scalar function, and given an index $i:I$, the __partial derivative__ $\partial_{i}$ is pointwise defined as $$\partial_{i}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x e_i) - f(v + y e_i)}{x - y}$$ ## See also ## * [[sequentially Cauchy complete Archimedean ordered field]] * [[real vector space]] * [[Newton-Leibniz operator]] * [[directional derivative]]