Definition:Differentiable Mapping/Real-Valued Function/Point

Definition
Let $\mathbb X$ be an open rectangle of $\R^n$.

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

Let $x = \left({x_1, x_2, \ldots, x_n}\right) \in \mathbb X$.

Then $f$ is differentiable at $x$ with respect to $x_i$ if:
 * $\displaystyle \lim_{h \to 0}{\frac{f \left({x_1, x_2, \ldots, x_i+h, \ldots, x_n}\right) - f \left({x_1, x_2, \ldots, x_i, \ldots, x_n}\right)}{h}}$

exists and is finite.

That is, if the partial derivative $\displaystyle\frac{\partial f}{\partial x_i}$ exists.

$f$ is described as differentiable at $x$ if $f$ is differentiable at $x$ with respect to $x_i$ for each $i=1, 2, \ldots, n$.

Also see

 * Characterization of Differentiability for clarification of this definition.