Derivative, partial derivative and gradient

derivative

The concept and application of derivative can be said to run through all our mathematical knowledge since junior high school . When the independent variable x And dependent variables y When both are one-dimensional and the definition field and value field are real number fields , The dependent variable y The definition of derivative is as follows ：

It's understandable , For a certain point （ The independent variables ）, When it changes dx When , Corresponding y（ The dependent variable ） The amount of change dy You can use the derivative f'(x) Work it out ：dy = f'(x) * dx.

The derivative of a point on a curve = The slope of the tangent passing through this point .

A concept that needs to be clarified is ： Although the derivative has positive and negative , It is still a scalar

Partial derivative

The partial derivative is when the dependent variable is one variable , When the independent variable is multivariate , The derivative of the dependent variable with respect to each independent variable . For example , The dependent variable of is z, The independent variable is x and y,z Yes x and y You can find the partial derivatives separately .

Actually , In the non strict sense, the derivative is unary “ Partial Guide ”, Take the derivative of only one independent variable .

gradient

Gradient is mainly for functions with multiple independent variables , Represents the direction in which the value of the function increases most rapidly , Visible gradient is a vector , It has direction .

In calculation , We need to find the partial derivative of the function , That is to find the derivative of each independent variable , This indicates the change of the function in the direction of each independent variable .

Here we need to specify that for unary functions , Understanding of derivatives and gradients .

First of all, give the conclusion ： When the argument is unary , The derivative value of a point on a function can be approximately understood as a gradient , The direction of the gradient is the sign direction of the derivative value , It must be the axis of the independent variable （ That's what we have here x Axis ） parallel .

for instance ： The expression of a unary function is , be The derivative of is expressed as , When x The value is 1 when , The derivative value is 2, Its symbol is positive , To follow x The positive direction of the axis is the direction in which the function value increases fastest , That is, the direction of the gradient is x Axis direction , The size is 2; When x The value is -1 The time derivative is -2, Its sign is negative , To follow x The opposite direction of the axis is the direction in which the function value increases fastest , That is, the direction of the gradient is x The axis is in the opposite direction , The size is also 2.

The reason why the derivative is approximate to the gradient here is because , Derivative is a scalar , And the gradient is a vector , In the process of explanation here , We take the sign of the derivative value as the direction of the gradient .