Definition:Partial Derivative/Real Analysis/Point/Definition 2

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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 $i$th partial derivative of $f$ at $a$ is the limit:

$\map {\dfrac {\partial f} {\partial x_i} } a = \ds \lim_{x_i \mathop \to a_i} \frac {\map f {a_1, a_2, \ldots, x_i, \ldots, a_n} - \map f a} {x_i - a}$