# Directional derivative

In multivariable calculus, there is  a notion of derivative, that is directional derivative.

Directional derivative is special derivative, follows a vector.

Definition. Let $U \subset \mathbb{R}^m$ and $U$ is open set. Let $f: U \to \mathbb{R}^n$. Suppose $a \in U$. Given $v \in \mathbb{R}^m$ with $v \ne 0$, the directional of $f$ at $a$ corresponding to $v$ is:

$Df_v(a) = \lim\limits_{t \to 0}\dfrac{f(a + tu) - f(a)}{t},$ if the limit exists.

Another definition: The directional derivative of $f$ at $a \in U$ corresponding to $v$ is $Df(a)\cdot v$.

Theorem. Two above definitions are equivalent.