Different Definitions of the Directional Derivative 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?
 A: The motivation is that some authors want to focus entirely on the direction for the construction of the directional derivative, so there is no extra "speed" taken into account. The limit definition,
$$D_v f(x)=\lim_{\epsilon\to0} \frac{f(x+\epsilon\, v)-f(x)}{\epsilon},$$
divides the difference of $f$ at $x$ and a point precisely $\epsilon$ units away by the length of the displacement, which is $\epsilon$. If you allow $v$ to be non-unit length, then the quotient will divide by a quantity (which is, again, $\epsilon$) that is proportionate but not equal to the length of the underlying displacement (which is $\epsilon\|v\|$ in size). This discrepancy means that the more generally defined directional derivative is more than just direction-dependent, it is also determined by the "size" associated to the direction.
A: The importance of the first definition is when you take the "derivative" of a function with respect to a vector field instead of just a simple vector.  When your space is not Euclidean, (for example a sphere) this distinction is very important, as there doesn't always exist a smooth vector field, defined on the tangent space of the surface, (more generally a differentiable manifold) that is always non-zero (see Hairy Ball theorem). This means that you can't just divide the vector field by it's norm at each point.  The distinction seems trivial because the underlying geometry of Euclidean space.
A: If you define $D_v|_p(f)=\frac{1}{|v|}\left<\nabla  f(p),v\right>$, you have that the directional derivative is independent to the magnitude of $v$ as everyone hope, but in this way the function $v\mapsto D_v|_p(\cdot)$ isn't linear, infact:
$$\frac{1}{|u+v|}\left<\nabla  f(p),u+v\right>\neq \frac{1}{|u|}\left<\nabla  f(p),u\right>+\frac{1}{|v|}\left<\nabla  f(p),v\right>$$
We decide to treat only unit vectors and to define $D_v|_p(f)=\left<\nabla  f(p),v\right>$  so as to preserve both the independence of the derivative to the magnitude of $v$  and the additivity of the above function.
In differential geometry, if $v$ is a general vector, the directional derivative is $$D_v|_p(f)=\left<\nabla  f(p),v\right>=\frac{d}{dt}\big|_{t=0}f(x+tv)$$
so we allow  the value of directional derivative to change with the magnitude of $v$.
