Understanding Gradients: How Directional Derivatives Reveal Maximum Change

The gradient is a vector that indicates the direction in which the directional derivative of a function attains its maximum value at a point; the function changes most rapidly in the direction of the gradient, with a rate equal to the magnitude of the gradient.

Consider a single‑variable function f(x) . The rate of change of f along the x -axis is the derivative df/dx .

For a bivariate function f(x, y) , as shown in the figure, there exists a rate of change in every direction within the plane.

The rate of change along the x -direction is ∂f/∂x , and along the y -direction is ∂f/∂y . More generally, the rate of change in an arbitrary direction u (a unit vector) is given by the dot product ∇f·u , where ∇f denotes the gradient.

Introducing the notation ∇f , called the gradient, we have that when the direction u aligns with ∇f , the directional derivative reaches its maximum; when it is opposite, the derivative reaches its minimum. Thus the function has the greatest rate of change in the direction of the gradient.

Reference

https://mp.weixin.qq.com/s/VqWKlA-05F5srSBcE5K6-Q