First of all, let's talk about directional derivatives. Like the partial derivative, the directional derivative is also the rate of change of a function in a specific direction, but the partial derivative is special in the X axis and the Y axis. The directional derivative changes in all directions are generally different, so which direction is the largest? Which direction is the smallest? For the convenience of research, there is a definition of gradient. Obviously, the gradient is actually a vector with the partial derivative of X as the abscissa and the partial derivative of Y as the ordinate, and the directional derivative is equal to this vector multiplied by the unit vector in the specified direction. According to cross product's definition, for a given function, its partial derivative is certain (at the same point, of course), so when the given direction is consistent with the gradient direction, it changes the fastest.

Generally speaking, the definition of gradient is to study the magnitude of directional derivative more conveniently.

(ps: those partial derivative formulas are not fun, otherwise they can be explained clearly! ! ! Ask for adoption, dear ...)