Higher partial derivatives

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $i, j \in \{1, \dots, n\}$. Suppose that the partial derivative $\partial f / \partial x_i$ exists at each point of $U$. The second partial derivative of $f$ with respect to $x_i$ and then $x_j$ at $a \in U$ is $$\frac{\partial^2 f}{\partial x_j \, \partial x_i}(a) = \frac{\partial}{\partial x_j}\left(\frac{\partial f}{\partial x_i}\right)(a),$$ when this partial derivative exists. It is also written $f_{x_i x_j}(a)$ or $\partial_j \partial_i f(a)$. For $i = j$, one writes $\partial^2 f / \partial x_i^2$. By induction, a partial derivative of order $k \ge 2$ is a partial derivative of a partial derivative of order $k - 1$: $\dfrac{\partial^k f}{\partial x_{i_k} \cdots \partial x_{i_1}} = \dfrac{\partial}{\partial x_{i_k}} \left( \dfrac{\partial^{k-1} f}{\partial x_{i_{k-1}} \cdots \partial x_{i_1}} \right)$.