Editing Directional derivative (section)

=== For differentiable functions ===

If the function ''f'' is [[Differentiable function#Differentiability in higher dimensions|differentiable]] at '''x''', then the directional derivative exists along any unit vector '''v''' at x, and one has

<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}</math>

where the <math>\nabla</math> on the right denotes the ''[[gradient]]'', <math>\cdot</math> is the [[dot product]] and '''v''' is a unit vector.<ref>If the dot product is undefined, the [[gradient]] is also undefined; however, for differentiable ''f'', the directional derivative is still defined, and a similar relation exists with the exterior derivative.</ref> This follows from defining a path <math>h(t) = x + tv</math> and using the definition of the derivative as a limit which can be calculated along this path to get:
<math display="block">\begin{align}
0
&=\lim_{t \to 0}\frac {f(x+tv)-f(x)-tDf(x)(v)} t \\
&=\lim_{t \to 0}\frac {f(x+tv)-f(x)} t - Df(x)(v) \\
&=\nabla_v f(x)-Df(x)(v).
\end{align}</math>

Intuitively, the directional derivative of ''f'' at a point '''x''' represents the [[derivative|rate of change]] of ''f'', in the direction of '''v'''.