I have seen several different starting points for definition the directional derivative of a function $f$ at a point $p$. Ultimately though, they can all be reduced to the equivalent definition via the gradient:
$$ D_v f(p) = \langle \nabla f(p), v \rangle $$
What is not clear though is why some texts only allow $v$ to be a unit vector and why other texts have no such restriction. If $v$ is not a unit vector one can always be produced by dividing $v$ by its norm; however, strictly speaking, the two definitions (one which requires a unit vector and one which doesn't) will differ by a scaling factor.
So, my question is, is there any reason to restrict the definition to a unit vector? What is the motivation for some texts to allow only unit vectors?