Directional derivative

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 $\partial f / \partial x_i$ 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 at $a$, the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$, see this statement.

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.