Calc III · §14.6 directional derivatives

A directional derivative is the slope in a chosen direction.

D_uf(a,b) = lim_h→0 [ f(a+hu₁, b+hu₂) − f(a,b) ] / h = ∇f · u

Stand at P, face the unit vector u = ⟨cos θ, sin θ⟩, and slice the surface with the vertical plane through that heading. The cut is a curve; D_uf is the slope of its tangent line at P — how fast the height climbs per step in direction u. It equals ∇f·u = |∇f|·cos(angle to the gradient): steepest along ∇f, zero along a level curve.

Hill view — slice the surface, read the tangent slope

Map view — your heading u vs the gradient

Slope vs. direction — D_uf as you swing θ around

Same point, many directions — each heading has its own slope. Biggest one is |∇f|. 🧭