Directional Derivatives

Definition

Suppose $f: \R ^n \to \R $. Given $a \in \R ^n$ and $\delta \in \R ^n$, if the limit

\[ \lim_{t \to 0} \frac{f(a + t\delta ) - f(a)}{t} \]

exists, then we say that $f$ is differentiable at $a$ in the direction $x$. We call the value of the limit the directional derivative of $f$ at $a$, in the direction $\delta $.

Notation

We denote the directional derivative of $f$ at $a$ by $D_xf(x)$.