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

Definition
Let $U\subset\R^n$ be an open set.

Let $f : U \to \R$ be a real-valued function.

Let $a = (a_1,\ldots,a_n)^\intercal \in U$.

Let $f$ be differentiable at $a$.

Let $i\in\{1,\ldots, n\}$.

The $i$th partial derivative of $f$ at $a$ is the limit:
 * $\dfrac{\partial f}{\partial x_i}(a) = \displaystyle \lim_{x_i \to a_i} \frac {f\left( a_1,\ldots, x_i, \ldots,a_n\right) - f\left(a\right)}{x_i - a}$