Gradient & Directional Derivative

Gradient vector and directional derivative of a surface

Use x and y with + − * / ^ and functions like sin, cos, tan, ln, log, sqrt, exp.

Any non-zero vector — it is normalized to a unit direction û.

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
2
Unit direction û
(1, 0)
f at the point
5
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.

Gradient & Directional DerivativeHeat map of f(x, y) near the point — brighter means larger f — with the gradient and chosen-direction arrows drawn from the point. ∇f = (2, 4). Directional derivative Dᵤf = 2.∇fû