Iterated directional derivatives measure repeated change along prescribed directions. A first [directional derivative](/page/Directional%20Derivative) asks how a function changes when its input is moved infinitesimally along one vector. An iterated directional derivative asks the next question: after measuring change in direction $v$, how does that new scalar or vector-valued function change in direction $u$? This concept is a local child of the broader [derivative](/page/Derivative): it keeps the directional viewpoint while moving toward second-order information such as the Hessian matrix, Taylor expansions, and second-order tests in multivariable calculus.

The order of the directions matters at the level of definition. Writing $D_uD_v f(a)$ means first form the directional derivative $D_v f$ near $a$, then differentiate that resulting function in direction $u$ at $a$. Under stronger smoothness hypotheses the two orders agree, but without those hypotheses the two iterated derivatives can differ or one of them can fail to exist. This is why the definition records both directions and the neighbourhood on which the first directional derivative is available.

## Definition

A single directional derivative at a point uses only values of $f$ on one line through that point. To form an iterated directional derivative, the first directional derivative $D_vf$ must be defined near the point so that it becomes a genuine nearby function before the second directional derivative is applied. Requiring $f$ to be differentiable on a neighbourhood is the clean standard hypothesis that guarantees this.

[definition: Iterated Directional Derivative] Let $U \subset \mathbb{R}^m$ be an [open set](/page/Open%20Set), let $a \in U$, let $u, v \in \mathbb{R}^m$, and let $f: U \to \mathbb{R}^n$ be a function. Suppose that $f$ is differentiable on an open neighbourhood $W \subset U$ of $a$, and write $D_vf(x):=Df_x(v)$ for $x \in W$. The iterated directional derivative of $f$ at $a$, first in direction $v$ and then in direction $u$, is \begin{align*} D_uD_v f(a) := D_u(D_v f)(a), \end{align*} provided the directional derivative on the right exists. [/definition]

The neighbourhood hypothesis is not decorative. It ensures that $x \mapsto D_vf(x)$ is available near $a$, so the second directional derivative is an ordinary directional derivative of that nearby function.

A first computation keeps the notation anchored. Linear functions have no second-order change, so they provide the baseline case against which curvature is measured.

[example: Linear Function] Let $T: \mathbb{R}^m \to \mathbb{R}^n$ be linear, let $b \in \mathbb{R}^n$, and define $f(x)=T(x)+b$. For $x,v \in \mathbb{R}^m$, the directional derivative in direction $v$ is computed from the defining difference quotient: \begin{align*} \frac{f(x+tv)-f(x)}{t}=\frac{T(x+tv)+b-(T(x)+b)}{t}. \end{align*} By linearity of $T$, $T(x+tv)=T(x)+tT(v)$, so \begin{align*} \frac{T(x+tv)+b-(T(x)+b)}{t}=\frac{tT(v)}{t}=T(v) \end{align*} for every $t \ne 0$. Taking the limit as $t \to 0$ gives \begin{align*} D_vf(x)=T(v). \end{align*} Thus the function $x \mapsto D_vf(x)$ is the constant function with value $T(v)$. For $a,u \in \mathbb{R}^m$, its directional derivative in direction $u$ is therefore \begin{align*} D_uD_vf(a)=\lim_{s \to 0}\frac{D_vf(a+su)-D_vf(a)}{s}. \end{align*} Since $D_vf(a+su)=T(v)$ and $D_vf(a)=T(v)$, the quotient is \begin{align*} \frac{T(v)-T(v)}{s}=0 \end{align*} for every $s \ne 0$, so \begin{align*} D_uD_vf(a)=0. \end{align*} This example shows that affine linear functions have first-order slope but no second-order directional variation. [/example]

## Related Definitions

Many second-order questions move only along one line: whether a curve bends upward, whether a critical point is stable in a chosen direction, or what the quadratic term of a one-variable slice should be. This motivates the special case where the two directions coincide.

[definition: Second Directional Derivative] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, let $v \in \mathbb{R}^m$, and let $f: U \to \mathbb{R}^n$ be a function. If $f$ is differentiable on an open neighbourhood $W \subset U$ of $a$ and $D_v(D_v f)(a)$ exists, then the second directional derivative of $f$ at $a$ in direction $v$ is \begin{align*} D_v^2 f(a) := D_vD_v f(a). \end{align*} [/definition]

This is not a [second derivative](/page/Second%20Derivative) with respect to a coordinate unless $v$ is a coordinate vector. It measures curvature along the affine line $a + tv$, and the length of $v$ affects the scale: replacing $v$ by $\lambda v$ scales $D_v^2 f(a)$ by $\lambda^2$ when the second derivative exists in the bilinear sense.

The definition of an iterated directional derivative allows ordered differentiation even when no full second derivative exists. To connect this ordered operation with the usual second-order calculus, we need a regularity condition saying that the total derivative itself varies differentiably with the base point. That condition is twice Frechet differentiability at a point.

[definition: Twice Frechet Differentiable at a Point] 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 $W \subset U$ of $a$. The function $f$ is twice Frechet differentiable at $a$ if the map \begin{align*} W \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n), \qquad x \mapsto Df_x \end{align*} is differentiable at $a$. [/definition]

Here $Df_x$ denotes the total derivative at $x$, a [linear map](/page/Linear%20Map), not its matrix representation. In this notation, $\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$ means the space of linear maps from $\mathbb{R}^m$ to $\mathbb{R}^n$. For scalar-valued functions, the matrix representation of the second derivative is the Hessian matrix once coordinates are chosen.

## Equivalent Characterisations

The definition by repeated directional differentiation is economical, but it hides the two-variable nature of second-order change. If $f$ has a genuine second derivative at $a$, then two directions enter through a bilinear map. We write $\mathrm{Bil}(\mathbb{R}^m \times \mathbb{R}^m, \mathbb{R}^n)$ for the space of bilinear maps that take an ordered pair of directions in $\mathbb{R}^m$ and return a vector in $\mathbb{R}^n$. The next formal statement identifies the second derivative with such a bilinear object, which is the structure needed to compare the two orders of directional differentiation.

[quotetheorem:9039]

This theorem is the bridge from the ordered definition to the symmetric bilinear object used in second-order calculus. For scalar-valued functions, computation usually happens in coordinates. The following formula turns the bilinear second derivative into an expression involving the gradient and the Hessian matrix, making directional notation compatible with standard multivariable calculus.

[quotetheorem:331]