Directional Derivative

Concept

For a function $f(x,y)$ and a vector in its input space $\vec{\textbf{v}}$, the directional derivative of $f$ along $\vec{\mathbf{v}}$ is the rate at which $f$ changes as input moves along the vector.

While it is represented by a number of symbols, $\nabla_{\vec{\textbf{v}}}$ will be used to denote a directional derivative along $\vec{\textbf{v}}$ for these notes.

They can be thought of as generalized partial derivatives - as a partial derivative w/ respect to $x$ tells us the amount a change in the input parallel to the $x$ axis affects the output of the function, while this directional derivative describes how a change in any direction (as opposed to parallel to an axis) affects the change in the output in the function.

\begin{example}{Singular Axis Directional Derivatives}{partials} A partial derivative w/ respect to $y$ can be thought of as a directional derivative along $\vec{\mathbf{v}} = \widehat{\mathbf{j}}$ (so $\frac{\partial f}{\partial y} = \nabla_{\widehat{\mathbf{j}}}$). \end{example}

Computation

Computing a directional derivative based on what we know so far is relatively simple.

Take the example vector $\vec{\mathbf{v}} = \left[\begin{matrix}2 \\ 3 \\ -1 \end{matrix}\right]$.

$\nabla_{\vec{\textbf{v}}} f = 2\frac{\partial f}{\partial x} + 3\frac{\partial f}{\partial y} + (-1)\frac{\partial f}{\partial z}$

This makes sense as $\frac{\partial f}{\partial x}$ is the amount a change in the output of the function with a small change in $x$, so a combination of each of the partial derivatives gives you change in the function for an arbitratry vector.

This also means that it can be computed via the gradient: $\nabla f \cdot \vec{\textbf{v}}$ as $\nabla f$ is a vector of each of the partial derivatives and $\vec{\textbf{v}}$ is a vector.

Sources

This describes basic directional derivatives nicely.