The [derivative](/page/Derivative) of a function is the first systematic answer to the question: near a point, which linear map best approximates the function? Many questions in analysis need the next layer of information. A linear approximation tells us instantaneous velocity, but it does not tell us whether that velocity is increasing, rotating, flattening, or changing direction. The second derivative measures the change of the derivative itself.

This idea is familiar for functions $f: U \subset \mathbb{R} \to \mathbb{R}$, where $f''(a)$ measures acceleration or concavity. In several variables the same idea survives, but the output is no longer merely a number or a matrix without context. Since $Df_a$ is a [linear map](/page/Linear%20Map), the derivative of $Df$ at $a$ is naturally a linear map whose values are linear maps. After the standard identification, it is a bilinear map in two direction variables.

Second derivatives are the local language behind Taylor Theorem, Hessian Matrix, second-order tests for extrema, [convex functions](/page/Convex%20Function), Newton's method, curvature computations, and second-order differential equations. The concept is therefore a child of the derivative: it differentiates the first-order approximation once more.

## Definition

The first derivative varies with the base point. If $f$ is differentiable near $a$, then nearby points $x$ have derivatives $Df_x$. The second derivative is introduced to measure whether this derivative-valued map has its own linear approximation at $a$.

[definition: Second Derivative] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable on an open neighbourhood $V \subset U$ of $a$. Write $\mathcal{L}(E,F)$ for the space of linear maps from a [vector space](/page/Vector%20Space) $E$ to a vector space $F$. The second derivative of $f$ at $a$ is the derivative at $a$ of the map $Df: V \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$, $x \mapsto Df_x$. When it exists, it is denoted $D^2f_a$ and satisfies \begin{align*} D^2f_a \in \mathcal{L}(\mathbb{R}^m, \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)). \end{align*} [/definition]

The target space in the definition is accurate but cumbersome. In computations, a second derivative should accept two direction vectors and return the second-order response of the function. This motivates the standard bilinear reading of the same object.

[definition: Bilinear Form of the Second Derivative] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ have a second derivative at $a$. The [bilinear form](/page/Bilinear%20Form) associated to $D^2f_a$ is the map $D^2f_a: \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}^n$ defined by \begin{align*} D^2f_a(h,k)=(D^2f_a(h))(k). \end{align*} [/definition]

Equivalently, $D^2f_a \in \mathrm{Bil}(\mathbb{R}^m \times \mathbb{R}^m,\mathbb{R}^n)$, where $\mathrm{Bil}(E \times E,F)$ denotes the space of bilinear maps from $E \times E$ to $F$.

Once this identification is made, analysts usually use the same symbol $D^2f_a$ for both the operator-valued linear map and the bilinear map. The next definition gives the name for the pointwise existence condition, because many arguments need to distinguish existence at a point from continuous existence on a whole [open set](/page/Open%20Set).

[definition: Twice Differentiable at a Point] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$. The map $f$ is twice differentiable at $a$ if there is an open neighbourhood $V \subset U$ of $a$ such that $f$ is differentiable on $V$ and the map $Df: V \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ is differentiable at $a$. [/definition]

This pointwise notion separates existence of a second derivative from any choice of coordinates or matrices. When the codomain is $\mathbb{R}$, many second-order questions ask how to store the bilinear map in standard coordinates, and that storage device is the Hessian matrix.

[definition: Hessian Matrix] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}$ be twice differentiable at $a$. The Hessian matrix of $f$ at $a$ is the matrix $Hf_a \in \mathbb{R}^{m \times m}$ representing the bilinear map $D^2f_a$ in the standard basis, with entries \begin{align*} (Hf_a)_{ij}=D^2f_a(e_i,e_j), \qquad 1 \le i,j \le m. \end{align*} [/definition]

With this convention, the second-order contribution in Taylor expansion is expressed by $D^2f_a(h,h)$, or in coordinates by $h^\top Hf_a h$. A Hessian entry is obtained by differentiating a coordinate derivative again, so the coordinate-level notion of a second partial derivative needs its own definition.

[definition: Second Partial Derivative] Let $U \subset \mathbb{R}^m$ be open, let $f: U \to \mathbb{R}^n$, and let $a \in U$. If the partial derivative $\partial_{x_j} f$ is defined on a neighbourhood of $a$ and $\partial_{x_i}(\partial_{x_j}f)(a)$ exists, then the second partial derivative of $f$ at $a$ in the $x_j$ direction followed by the $x_i$ direction is \begin{align*} \partial_{x_i}\partial_{x_j} f(a):=\partial_{x_i}(\partial_{x_j}f)(a). \end{align*} [/definition]

The order of indices matters in the notation, and isolated second partial derivatives do not by themselves provide a stable second-order calculus. To make coordinate formulas, symmetry, and Taylor remainders behave well on open sets, one imposes continuity of the total second derivative.

[definition: Twice Continuously Differentiable Map] Let $U \subset \mathbb{R}^m$ be open and let $f: U \to \mathbb{R}^n$. The map $f$ is twice continuously differentiable on $U$, written $f \in C^2(U;\mathbb{R}^n)$, if $f$ is twice differentiable at every point of $U$ and the map $D^2f: U \to \mathcal{L}(\mathbb{R}^m, \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n))$, $x \mapsto D^2f_x$, is continuous. [/definition]

This continuity assumption is stronger than pointwise existence. It is the hypothesis that makes mixed partial derivatives interchangeable and makes second-order Taylor remainders stable near the point.

## Equivalent Characterisations

The most useful alternative view of the second derivative is through a second-order approximation. The first derivative gives an affine approximation; the second derivative gives the next correction term. This perspective explains why the second derivative is bilinear: a second-order correction is quadratic in the displacement. In the Taylor remainder below, $o(|h|^2)$ means a term whose norm divided by $|h|^2$ tends to $0$ as $h \to 0$; it is smaller than every fixed quadratic error near the point.