\begin{definition}[Definition:Partial Derivative/Real Analysis/Point/Definition 1]
Let $U \subset \R^n$ be an open set.
Let $f: U \to \R$ be a real-valued function.
Let $a = \tuple {a_1, a_2, \ldots, a_n}^\intercal \in U$.
Let $f$ be differentiable at $a$.
Let $i \in \set {1, 2, \ldots, n}$.
The '''partial derivative of $f$ with respect to $x_i$ at $a$''' is denoted and defined as:
:$\map {\dfrac {\partial f} {\partial x_i} } a := \map {g_i'} {a_i}$
where:
:$g_i$ is the real function defined as $\map g {x_i} = \map f {a_1, \ldots, x_i, \dots, a_n}$
:$\map {g_i'} {a_i}$ is the derivative of $g$ at $a_i$.
Category:Definitions/Partial Differentiation
\end{definition}