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.

A unit direction from x0 in the input plane and the graph of h(t)=f(x0+tv), whose tangent at t=0 gives the directional derivative.

For a scalar-valued function, set $h(t)=f(\boldsymbol{x}_0+t\boldsymbol{v})$ $t v)$ . The directional derivative is the right-hand slope $h'_+(0)$ . In the figure, $f(x_1,x_2)=x_1^2+x_2^2$ $x_{2}^{2}$ , $\boldsymbol{x}_0=(1,0)$ and $\boldsymbol{v}=(3/5,4/5)$ $(3/5, 4/5)$ , so $h(t)=1+6t/5+t^2$ and the slope is $6/5$ . The two panels connect movement in the input plane with the resulting one-variable curve. The smooth examples in the interactive figures have the same slope from either side; the definition above only asks for the limit from $t>0$ $0$ .

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: real numbers, subset

Last updated: 27 October 2026 11:52