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 $k - 1$: $\dfrac{\partial^k f}{\partial x_{i_k} \cdots \partial x_{i_1}} = \dfrac{\partial}{\partial x_{i_k}} \left( \dfrac{\partial^{k-1} f}{\partial x_{i_{k-1}} \cdots \partial x_{i_1}} \right)$.

Remarks

In $\partial^2 f / \partial x_j \, \partial x_i$ the variable nearest to $f$ is differentiated first. In $f_{x_i x_j}$ the variable written first is differentiated first. For $f(x, y) = x^3 y^2$, one gets $f_{xx} = 6 x y^2$, $f_{xy} = f_{yx} = 6 x^2 y$ and $f_{yy} = 2 x^3$. If the second partial derivatives are continuous, then the order of differentiation does not matter, by Schwarz's theorem. Without continuity, the mixed partial derivatives can differ. The second partial derivatives at a point form the Hessian matrix.

Explanations

L1 human

A second partial derivative measures how a slope changes. For example, $f_{xy}$ measures how the slope in the $x$-direction changes as you move in the $y$-direction. You compute it by differentiating twice, one variable at a time.