why normalize and the definition of directional derivative I'm not understanding how to solve this problem.
I think the problem lies in the fact that I don't understand why normalize and the definition of directional derivative...
It's in Portuguese. The question would be: Determine the directional derivative of f in P in direction of vector u:

 A: The directional derivative of a scalar function $f:\mathbb{R}^n\to\mathbb{R}$ at a point $P$ in the direction of a vector $\mathbf{u}$ is usually (but not always) defined as $$\nabla f_P\cdot\frac{\mathbf{u}}{\|\mathbf{u}\|}$$
where $\nabla f_P$ is the gradient of $f$ evaluated at $P$.
The normalization in this definition is motivated for at least two reasons:


*

*Normalizing allows you to interpret the directional derivative as the rate of change of the function per unit distance in the direction of $\mathbf{u}$.

*You can't meaningfully compare the rates of change of the function in different directions unless you use vectors of the same length. So why not use vectors of unit length? That correctly captures the fact that a directional derivative should depend only on the direction of the vector, not on its magnitude.
EDIT
There is a nice footnote on the issue in Hubbard's book:

To compare derivatives in different directions one must first normalize the vectors to have the same length. If $\mathbf{v_1}$ is a kilometer long, $\mathbf{v_2}$ is a centimeter long, and "time 1" is a minute, then $\mathbf{x}$ is traveling 60 kilometers an hour in the direction $\mathbf{v_1}$ compared to 60 centimeters an hour in direction $\mathbf{v_2}$; knowing how much $\mathbf{f(x)}$ varies when $\mathbf{x}$ travels one minute in these two directions does not tell you in which direction $\mathbf{f(x)}$ is varying fastest at time 0 (i.e., as $h\to0$). For this reason, some authors allow only vectors $\mathbf{v}$ of length 1 to be used to define directional derivatives. We feel this restriction is undesirable, as it loses the essential linear character of the directional derivative as a function of $\mathbf{v}$.

