Definition:Partial Derivative/Real Analysis/Point/Definition 1
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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 $a_i$.