Partial derivative

Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $i \in \{1, \dots, n\}$. Let $e_i$ be the $i$-th standard basis vector of Euclidean space $\mathbb{R}^n$. The partial derivative of $f$ with respect to $x_i$ at $a$ is the limit $$\frac{\partial f}{\partial x_i}(a) = \lim_{h \to 0} \frac{f(a + h e_i) - f(a)}{h},$$ when this limit exists in $\mathbb{R}$. Other notations for it are $\partial_i f(a)$, $D_i f(a)$ and $f_{x_i}(a)$.