Directional Derivative¶

Definition¶

Consider a subset $E \subseteq \mathbb{R}^n$ and a function $\boldsymbol{f}\colon E\to\mathbb{R}^k$ $R^{k}$ . Let $\boldsymbol{x}_0$ be an interior point of $E$ and $\boldsymbol{v}$ be a unit vector in $\mathbb{R}^n$ (note: some authors only require $\boldsymbol{v}$ to be a nonzero vector). If the limit

\lim_{t\to 0^+}\frac{\boldsymbol{f}(\boldsymbol{x}_0+t\boldsymbol{v})-\boldsymbol{f}(\boldsymbol{x}_0)}{t}

exists, then $\boldsymbol{f}$ is said to be differentiable in the direction $\boldsymbol{v}$ at $\boldsymbol{x}=\boldsymbol{x}_0$ $x_{0}$ . This limit is termed the directional derivative and is denoted by $\mathrm{D}_{\boldsymbol{v}} \boldsymbol{f}(\boldsymbol{x}_0)$ .

Here unit length is measured by the Euclidean Norm, $\|\boldsymbol{v}\|_2=1$ $1$ . Equivalently, the Inner Product satisfies $\langle\boldsymbol{v},\boldsymbol{v}\rangle=1$ $1$ , so $t$ measures distance travelled along the chosen direction.

Interpretation¶

The directional derivative measures the instantaneous rate of change of a function as you move from a point in a specific direction.

It generalizes the one-sided derivative to functions of several variables by looking at how the function changes along a straight line in that direction.

Elliptical level curves of f with the gradient at a point x0 pointing outwards and a unit direction v at an angle to it; the directional derivative is the gradient dotted with v.

For a differentiable $f$ and a unit direction $\boldsymbol{v}$ , the directional derivative is $D_{\boldsymbol{v}}f(\boldsymbol{x}_0)=\boldsymbol{\nabla} f(\boldsymbol{x}_0)\cdot\boldsymbol{v}$ $v$ . For the nonzero gradient drawn, it is largest when the direction follows the gradient and zero in a tangent direction to the level curve.

Turn the direction $\boldsymbol v$ with $\theta$ : the slope of $f$ along $\boldsymbol v$ is $\boldsymbol{\nabla} f(\boldsymbol x_0)\cdot\boldsymbol v$ $v$ , largest along the gradient and zero along the level curve.

Turn $\boldsymbol v$ : the slice of the graph through $\boldsymbol x_0$ in direction $\boldsymbol v$ has slope $\mathrm{D}_{\boldsymbol v}f(\boldsymbol x_0)=\boldsymbol{\nabla} f(\boldsymbol x_0)\cdot\boldsymbol v$ $v$ .

Key Results¶

If $\boldsymbol{f}$ is totally differentiable at $\boldsymbol{x}_0\text{,}$ then the directional derivative $\mathrm{D}_{\boldsymbol{v}}\boldsymbol{f}(\boldsymbol{x}_0)$ exists for every direction $\boldsymbol{v}$ and equals the total derivative at $\boldsymbol{v}\text{,}$ $\mathrm{D} \boldsymbol{f}_{\boldsymbol{x}_0}(\boldsymbol{v})\text{.}$
A function is totally differentiable at a point implies that this function is directionally differentiable in any direction. The converse is incorrect.

Examples¶

For the function $\boldsymbol{f}(x,y)=(x^2,y^2)$ $(x^{2}, y^{2})$ , consider $\boldsymbol{x}_0=(1,2)$ and $\boldsymbol{v}=(3/5,4/5)$ $(3/5, 4/5)$ . We have $\displaystyle \mathrm{D}_{\boldsymbol{v}}\boldsymbol{f}(\boldsymbol{x}_0) =\left(\frac{6}{5},\frac{16}{5}\right)$ $(\frac{6}{5}, \frac{16}{5})$ .

Prerequisites¶

Total Derivative as a Linear Map

Sources¶

Math III: Analysis · Lecture 11

Symbols: nabla gradient, real numbers, subset, theta

Last updated: 17 October 2026 21:53