Gradient & Directional Derivative

Gradient vector and directional derivative of a surface

f(x, y) = x^2 + y^2 Dᵤf = ∇f · û

Gradient ∇f

(2, 4)

Magnitude |∇f|

4.472135955

The greatest rate of increase — the slope in the steepest-ascent direction.

Directional derivative Dᵤf

Unit direction û

(1, 0)

f at the point

Steepest-ascent angle

63.43494882°

Angle between ∇f and û

63.43494882°

Dᵤf = ∇f · û = |∇f|·cos θ — the projection of the gradient onto your direction.