Directional derivative
High school
Consider a function of several variables, like the height of a hill at each point of a map. The directional derivative of the function is its vertical rate of change as you move horizontally in a particular direction.
Undergraduate
Consider a function $f$, a point $x$, and a direction $v$. The directional derivative of $f$ at $x$ in the direction $v$ is the rate of change of $f$ in that direction:
\[D_v f(x) = \lim_{t \to 0} \frac{f(x + t v) - f(x)}{t}.\]Examples
Worked examples
-
$f(x, y) = x^2 + 3y$
: $D_v f(x, y) = \nabla f(x, y) \cdot v = 2x$, in the direction $v = (1, 0)$
Undergraduate
$\nabla f \cdot (1, 0) = 2x$.
Properties
- If $f$ is differentiable, then $D_v f(x) = \nabla f(x) \cdot v$.
- A function can have directional derivatives in every direction and still not be differentiable, or even continuous.
Notes
Some authors require $v$ to be a unit vector, so that $D_v f(x)$ is a rate per unit distance.
Distinctions involving this term
- Types of multivariable derivatives: All of these package the partial derivatives of a function differently: as a vector (gradient), a matrix (Jacobian), a linear map (differential), or a single number in a chosen direction (directional derivative) or along a vector field (Lie derivative).