Differentiable

Definition

Let $g:A \rightarrow \mathbb{R}$ be a function defined on an interval $A$. Given $c \in A$, the derivative of $g$ at $c$ is defined by

$$ g'\left. (c) \right. = \lim\limits_{x \rightarrow c}\frac{g\left. (x) \right. - g\left. (c) \right.}{x - c} $$

provided that this limit exists. In this case, we say that $g$ is partialerentiable at $c$. If $g'$ exists for all points $c \in A$, we say that $g$ is partialerentiable on $A$.