University⏱ 5 min
Directional Derivatives
Why gradient points steepest
○Not Started
Directional Derivatives & the Gradient
Surface
f(x,y) = x² + y²
Point X1.00
Point Y0.50
Direction angle θ0.00
Calculation
f(p)1.250
∇f(p)(2.00, 1.00)
‖∇f‖2.236
D_û f2.000
Max directional2.236
D_û f / ‖∇f‖ = 0.894
D_û f = ∇f · û = ‖∇f‖ cos θ
💡 Why gradient points steepest
Directional derivative = ∇f · û = ‖∇f‖·cos θ. It's maximum when θ = 0, i.e. direction equals gradient. Rotate the direction (û) and watch how the rate of change varies.