Editing Directional derivative (section)

=== Using only direction of vector ===
[[image:Geometrical interpretation of a directional derivative.svg|thumb|The angle ''α'' between the tangent ''A'' and the horizontal will be maximum if the cutting plane contains the direction of the gradient ''A''.]]
In a [[Euclidean space]], some authors<ref>Thomas, George B. Jr.; and Finney, Ross L. (1979) ''Calculus and Analytic Geometry'', Addison-Wesley Publ. Co., fifth edition, p. 593.</ref> define the directional derivative to be with respect to an arbitrary nonzero vector '''v''' after [[Normalized vector|normalization]], thus being independent of its magnitude and depending only on its direction.<ref>This typically assumes a [[Euclidean space]] – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.</ref>

This definition gives the rate of increase of {{math|''f''}} per unit of distance moved in the direction given by {{math|'''v'''}}. In this case, one has
<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h|\mathbf{v}|}},</math>
or in case ''f'' is differentiable at '''x''',
<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} .</math>