Definition. differentiation [br911]

Suppose $E$ is an open set in $\mathbb{R}^{n}$, $\mathbf{f}$ maps $E$ into $\mathbb{R}^{m}$, and $\mathbf{x} \in E$. If there exists a linear transformation $A$ of $\mathbb{R}^{n}$ into $\mathbb{R}^{m}$ such that $$ \lim_{ \mathbf{h} \to 0 } \frac{\lvert \mathbf{f}(\mathbf{x} + \mathbf{h}) - \mathbf{f}(\mathbf{x}) - A \mathbf{h} \rvert }{\lvert \mathbf{h} \rvert } = 0, $$ then we say $\mathbf{f}$ is differentiable at $\mathbf{x}$, and we write $$ \mathbf{f}'(\mathbf{x}) = A. $$ If $\mathbf{f}$ is differentiable at every $\mathbf{x} \in E$, we say that $\mathbf{f}$ is differentiable in $E$.