Math basics › Multivariable calculus
Gradients and directional derivatives
The gradient
∇f = ⟨∂f/∂x, ∂f/∂y⟩, with ∂f/∂z added in 3D. A vector made of all the partial derivatives.
What it points at
∇f points the way f increases fastest, and its length is that fastest rate. It is perpendicular to the level curves.
Directional derivative
The rate of change in the direction of u is ∇f · u, and u must be a unit vector. Divide it by its length first.