Mathematical knowledge - Derivation - Basic derivation knowledge

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Preface

Mathematical derivation in the field of machine learning and deep learning always uses a lot of theoretical knowledge of derivation , The frequency of occurrence is very high . So it is very important to gather some knowledge about derivation .

One 、 Definition of derivative

derivative （Derivative）, It's also called derivative value . Also known as wechat business , It is an important basic concept in calculus . Function as y=f（x） The independent variable of x In a bit x0 Generate an increment on Δx when , The increment of the output value of the function Δy And the independent variable increment Δx The ratio is Δx Tend to 0 The limit of time a If there is ,a That is in x0 Derivative at , Write it down as f’（x0） or df（x0）/dx.

Derivative is the local property of a function . The derivative of a function at a point describes the rate of change of the function near that point . If the arguments and values of the function are real numbers , The derivative of a function at a certain point is the tangent slope of the curve represented by the function at this point . The essence of derivative is the local linear approximation of function through the concept of limit . For example, in kinematics , The derivative of the displacement of an object with respect to time is the instantaneous velocity of the object .

Not all functions have derivatives , A function does not necessarily have derivatives at all points . If the derivative of a function exists at a certain point , It is said to be differentiable at this point , Otherwise, it is called non differentiable . However , A differentiable function must be continuous ; Discontinuous functions must not be differentiable .

For differentiable functions f(x),x↦f’(x) It's also a function , Referred to as f(x) The derivative of （ Derivative for short ）. The process of finding the derivative of a known function at a certain point or its derivative function is called derivation . In essence , Derivation is a process of finding limits , The four algorithms of derivative also come from the four algorithms of limit . conversely , Given the derivative function, you can also find the original function in reverse , Indefinite integral .

The basic theorem of calculus shows that finding the original function is equivalent to integral . Derivation and integration are a pair of reciprocal operations , They are the most basic concepts in calculus .[1]

Two 、 Derivatives of elementary functions

3、 ... and 、 The operation of complex derivatives

1、 Four operations of derivative ：

2、 Derivative relation between original function and inverse function （ The derivative of trigonometric function is used to deduce the derivative of trigonometric function ）：

y=f（x） The inverse of is x=g（y）, Then there are y’=1/x’.

3、 The derivation method of derivative of compound function ：

The derivative of a compound function with respect to an independent variable , Equal to the derivative of a known function over an intermediate variable , Multiply by the derivative of the intermediate variable over the independent variable （ It's called the chain rule ）.

4、 The derivation rule of variable limit integral ：

（a（x）,b（x） Is a subfunction ）

Four 、 Expansion of matrix derivation

Mathematical knowledge - Matrix article - matrix / Vector derivation

May refer to

[1] Baidu Encyclopedia ： The basic concept of derivative