Derivative, partial derivative, directional derivative

1. Derivative and partial derivative

1.1. derivative

Derivative defines ： The reaction is a function $y=f(x)$ At some point along the argument $x$ In the right direction （ namely : $x$ Affirmative direction ） The rate of change of .

Derivative formula ：

function $y=f(x)$ stay $x_0$ The derivative of a point is written as $f^{\prime }(x_0)$, be $f^{\prime }(x_0)$ by ：
$$f^{\prime }(x_0)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$$

Geometric meaning ： function $y=f(x)$ stay $x_0$ The derivative of a point $f^{\prime }(x_0)$ It means that the function curve is at the point $P_0(x_0,f(x_0))$ The slope of the tangent at 【 The geometric meaning of the derivative is that the function curve is at this point $P_0(x_0,f(x_0))$ The tangent slope on the 】.

1.2. Partial derivative

Partial derivative definition ： Take the binary function for example , The reaction is a function $z=f(x,y)$ At a certain point along a positive axis （ namely ： Along $x$ The axis is in the positive direction or along $y$ Affirmative direction ） The rate of change of .

Partial derivative formula ：

In binary function $z=f(x,y)$ For example ：
function $z=f(x,y)$ stay $(x_0,y_0)$ Point to point pair $x$ The partial derivative of is written as $\frac{\partial z}{\partial x}$（ It can also be recorded as ： $\frac{\partial f}{\partial x}$ , $z_x$ or $f_x(x,y)$）, be $\frac{\partial z}{\partial x}$ by ：

$$\frac{\partial z}{\partial x}=\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$$

function $z=f(x,y)$ stay $(x_0,y_0)$ Point to point pair $y$ The partial derivative of is written as $\frac{\partial z}{\partial y}$（ It can also be recorded as ： $\frac{\partial f}{\partial y}$ , $z_y$ or $f_y(x,y)$）, be $\frac{\partial z}{\partial y}$ by ：

$$\frac{\partial z}{\partial y}=\underset{\Delta y\to 0}{\lim }\frac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}$$

notes ： The essence of derivative and partial derivative is ⼀ To , All when ⾃ The change of variables tends to 0 when , The change of function value is related to ⾃ Changes in variables , Between them ⽐ Limit of value .

3. Directional derivative

before ⾯ derivative and Partial derivative The definition of , All along The coordinate axis is positive ⽅ towards Discuss the rate of change of the function . So when we talk about functions Arbitrarily along ⽅ Rate of change in direction when , That leads to ⽅ Definition of derivative .

Directional derivative ： The reaction is a function $y$ At some point $x_0$ Along a specific direction （ Is not necessarily $x$ The axis is in the right direction ） The rate of change of .