Why the Gradient Points to the Steepest Ascent: Understanding Directional Derivatives

The gradient is a vector that indicates the direction in which a function increases most rapidly at a given point, and its magnitude equals the maximal rate of change.

For a single‑variable function f(x) , the derivative f'(x) gives the rate of change along the x‑axis.

For a bivariate function f(x, y) , as illustrated below, the rate of change can be examined along any direction in the plane.

The directional derivative of f in a unit direction u = (u_x, u_y) is D_u f = ∇f \cdot u , where ∇f = (∂f/∂x, ∂f/∂y) is the gradient.

Introducing the notation ∇f for the gradient, we have

∇f = (∂f/∂x, ∂f/∂y)

When the direction u aligns with the gradient ( u = ∇f / \|∇f\| ), the directional derivative attains its maximum value \|∇f\| ; when u points opposite to the gradient, the derivative reaches its minimum -\|∇f\| . Thus, moving along the gradient yields the steepest ascent of the function.