1. The Question Generalised — Infinitely Many Roads
Callback — Partial Derivatives as Two Coordinate Cuts
When we studied partial derivatives, we learned to measure the rate of change of $f(x,y)$ in exactly two directions: along the $x$-axis (holding $y$ fixed) and along the $y$-axis (holding $x$ fixed). Those two coordinate cuts gave us $f_x$ and $f_y$. But standing at a point on a surface, you can walk in infinitely many directions — not just east or north, but northeast, south-southwest, or any bearing you choose. The partial derivatives are just two roads among infinitely many.
The directional derivative answers the natural next question: what is the rate of change of $f$ along an arbitrary direction?
The Direction as a Unit Vector — A Non-Negotiable Law
A direction in the plane (or in space) is specified by a vector. But we need a unit vector $\mathbf{u}$ — a vector of length 1. Why?
Given any nonzero vector $\mathbf{v}$, the corresponding unit vector is:
$$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$$