The gradient ∇f — a compass pointing uphill.

∇f(x, y) = ⟨ ∂f/∂x , ∂f/∂y ⟩

• Bundle the two partials into one vector. It lives in the input (x, y) plane.
• It points in the direction of steepest increase of f, and its length |∇f| is that fastest rate of climb.
• It is perpendicular to the level curve f = constant at that point.
• Rate of change in a chosen unit direction u is the directional derivative Duf = ∇f · u — biggest when u lines up with ∇f, zero along the level curve.