---
tag: 05EGFBZTGC
kind: definition
name: Having a directional derivative
msc: [26B05]
defines: 06KWGHDN2J
uses: [0XPR1E85W0]
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 $a \in U$, let $v \in \mathbb{R}^n$, and let $L \in \mathbb{R}^m$.
Let $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$.
Equivalently, $\dfrac{f(a + t v) - f(a)}{t} \to L$ as $t \to 0$.

## Remarks

This definition states a property of $f$, $a$, $v$ and $L$. The [[directional derivative|0Y7PVW4Q44]] $D_v f(a)$ is then the vector $L$, which is unique.
Only the values of $f$ on the line through $a$ in the direction $v$ matter. So $f$ can have a directional derivative in each direction at $a$ and still fail to be continuous at $a$, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

The function $g(t) = f(a + t v)$ is the function $f$ seen along the straight line through $a$ with direction $v$.
To have a directional derivative means that this function of one variable has a derivative at the start of the line.