Definition:Differential/Real-Valued Function

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At a point

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

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

Let $f$ be differentiable at a point $x \in U$.

$\displaystyle \d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac {\partial f \left({x}\right)} {\partial x_i} h_i = \frac {\partial f \left({x}\right)} {\partial x_1} h_1 + \frac {\partial f \left({x}\right)} {\partial x_2} h_2 + \cdots + \frac {\partial f \left({x}\right)} {\partial x_n} h_n$


$h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$
$\dfrac {\partial f} {\partial x_i}$ is the partial derivative of $f$ with respect to $x_i$.


Substituting $\rd x_i \left({x; h}\right) = h_i$ for $i = 1, 2, \ldots, n$, the following notation emerges:

$\displaystyle \rd f := \sum_{i \mathop = 1}^n \frac {\partial f} {\partial x_i} \rd x_i = \frac {\partial f} {\partial x_1} \rd x_1 + \frac {\partial f} {\partial x_2} \rd x_2 + \cdots + \frac {\partial f} {\partial x_n} \rd x_n$