Why is the negative gradient direction the direction of local steepest descent?

1 First we need to know some mathematical concepts:

1) Derivative
When the function definition domain and value are in the real number domain, the derivative can represent the tangent slope on the function curve. In addition to the slope of the tangent, the derivative also represents the rate of change of the function at that point.

Pay attention to the unary function, only one independent variable is changing, that is, there is only one direction of change rate. When it comes to multivariate directions, partial derivatives must be introduced.

2) Partial derivative The
partial derivative refers to the rate of change of the multivariate function along the coordinate axis. For the partial derivative of the binary function, it can be fixed according to a certain variable (as a constant), and only the other variable can be calculated. Rate of change. From the animation below, you can see that the partial derivative of x is obtained, the section is parallel to the xOz plane, and the intersection of the section and the binary function is a curve of a one-variable function.

3) How does the directional derivative
solve the rate of change in any direction? That is the directional derivative. The direction in which the directional derivative takes the maximum value is the gradient, which is the direction with the largest rate of change of the function. Observe that the directional derivative (blue) and the The arrows of the gradient (black) point to (both here only indicate direction).
Suppose$z=f(x,y)$is a surface,$p(x,y)$is a point in the domain of f, then for the slope of the vector u direction, we can use partial differentiation to simplify the calculation. Finally, with the help of the definition of vector dot product, we can conclude that when the angle between vector A and vector I is 0, the rate of change of the function in this direction is the largest.