---
tag: 0Y7PVW4Q44
kind: definition
name: Directional derivative
msc: [26B05]
defines: 06KWGHDN2J
uses: [0XPR1E85W0, 09EGC6F8PT]
sources:
  - url: https://en.wikipedia.org/wiki/Directional_derivative
    locator: Definition
---

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, and let $v \in \mathbb{R}^n$.
The directional derivative of $f$ at $a$ in the direction $v$ is the limit
$$D_v f(a) = \lim_{t \to 0} \frac{f(a + t v) - f(a)}{t},$$
when this limit exists in $\mathbb{R}^m$.
Other notations for it are $\partial_v f(a)$ and $\nabla_v f(a)$.

## Remarks

For $v = e_i$, the directional derivative is the [[partial derivative|09EGC6F8PT]] $\partial f / \partial x_i$ at $a$.
Some books require that $v$ is a unit vector, so that $D_v f(a)$ is a rate of change per unit of distance.
If $D_v f(a)$ exists and $c \in \mathbb{R}$, then $D_{cv} f(a)$ exists and equals $c \, D_v f(a)$.
When $f$ is [[differentiable|0HHHDCXZV4]] at $a$, the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$, see [[this statement|0A6PMR20N4]].

## Explanations

### L1 human

Stand at the point $a$ on the graph of $f$ and walk in the direction $v$.
The directional derivative is the slope that you feel at the first step.
For $f(x, y) = x^2 + y^2$ at $a = (1, 0)$ and $v = (0, 1)$, you get $f(1, t) = 1 + t^2$, so the slope is $0$. You start along the tangent of the circle $x^2 + y^2 = 1$, where $f$ is constant.