The directional derivative of a multivariate differentiable function along a given vector $\mathbf{v}$ at a given point $\mathbf{x}$ intuitively represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.

Definition

The directional derivative of a scalar function $f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n)$ along a vector $\mathbf{v} = (v_1, \ldots, v_n)$ is the function defined by the limit:

$\displaystyle{\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}}$

If the function $f$ is differentiable at $\mathbf x$, then the directional derivative exists along any vector $\mathbf v$, and one has $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}$, where the $\nabla f(\mathbf{x})$ is the gradient vector and $\cdot$ is the dot product. At any point $\mathbf x$, the directional derivative of $f$ intuitively represents the rate of change of $f$ with respect to time when it is moving at a speed and direction given by $\mathbf v$ at the point $\mathbf x$. The name "directional derivative" is a bit misleading since it depends on both the length and direction of $\mathbf v$.

We can imagine the directional derivative $\nabla_{\mathbf{v}}{f}(\mathbf{x})$ as the slope of the tangent line to the 2-dimensional slice of the graph of $f$ that lies parallel to the vector $\mathbf{v}$. However, this slice will be stretched or compressed horizontally unless $\mathbf{v}=1$.

Gradient of a Function

The gradient of the function $f(x,y) = −\left((\cos x)^2 + (\cos y)^2\right)$ depicted as a projected vector field on the bottom plane. Directional derivative represents the rate of change of the function along any direction specified by $\mathbf{v}$.

Properties

Many of the familiar properties of the ordinary derivative hold for the directional derivative.

The Sum Rule

$\nabla_\mathbf{v} (f + g) = \nabla_\mathbf{v} f + \nabla_\mathbf{v} g$

The Constant Factor Rule

For any constant $c$, $\nabla_\mathbf{v} (cf) = c\nabla_\mathbf{v} f$.

The Product Rule (or Leibniz Rule)

$\nabla_\mathbf{v} (fg) = g\nabla_\mathbf{v} f + f\nabla_\mathbf{v} g$

The Chain Rule

If $g$ is differentiable at $p$ and $h$ is differentiable at $g(p)$, then $\nabla_\mathbf{v} h\circ g (p) = h'(g(p)) \nabla_\mathbf{v} g (p)$.