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

Let $f$ be differentiable at $a$.

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

The partial derivative of $f$ with respect to $x_i$ at $a$ is denoted and defined as:


 * $\dfrac {\partial f}{\partial x_i}(a) := g_i'(a_i)$

where:
 * $g_i$ is the real function defined as $g \left({x_i}\right) = f \left({a_1, \ldots, x_i, \dots, a_n}\right)$
 * $g'(a_i)$ is the derivative of $g$ at $a_i$.