Definition:Partial Derivative/Real Analysis/Point/Definition 1

Definition
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 $a_i$.