Multivariate calculus from first principles
Partial derivatives only tell you the slope along the coordinate axes, but you can walk off in any direction. The directional derivative D_u f answers: if I step along the unit vector u, how fast does f change? The answer turns out to be a single dot product with the gradient.
Imagine hiking across that same hill, but instead of facing straight uphill you pick a compass bearing, say north-east, and walk that way. The directional derivative D_u f is the slope you actually feel under your boots along that heading. Head toward the steepest direction and you feel the full climb; turn sideways along the hillside and the ground feels flat.
Since D_u f = ∇f·u = ‖∇f‖‖u‖cos θ = ‖∇f‖cos θ (because u is a unit vector), the rate of change is largest exactly when cos θ = 1, that is, when u points along ∇f. Spin the direction arrow below and watch the slope readout peak when it aligns with the gradient and vanish when it's perpendicular.