【开篇语】

微分概念在整个微积分体系中占有重要地位。理解微分概念是微积分教育的重要环节。在历史上，微分的定义经历了很长时间的发展。牛顿、莱布尼兹是微积分的主要创建人，他们的微积分可以称为第一代微积分，第一代微积分的方法是没有问题的，而且获得了巨大的成功，但是对微分的定义（即微分的本质到底是什么）的说明不够清楚；以柯西、维尔斯特拉斯等为代表的数学家在极限理论的基础上建立了微积分原理，可以称之为第二代微积分，并构成当前教学中微积分教材的主要内容。第二代微积分与第一代微积分在具体计算方法上基本相同，第二代微积分表面上解决了微分定义的说明，但是概念和推理繁琐迂回。

当前，围绕微分定义问题，国内外学术界已经开始形成一些讨论，参与者从科学院院士，中青年数学工作者，以致在读博士硕士，当然也包括一些毫无话语权的“N无数学家”。但真理面前人人平等，只要我们抱着持之有故言之成理的科学态度，相信会引发深刻的思考。

为了使得微分定义的讨论更加深入，并且有充足的养料支撑，有必要将古今中外现行微积分学术著作中的微分定义详细调查。从今天起，我将在我所搜集整理的微积分定义逐次摘录在网上，方便大家讨论。在摘录的同时，将做一些简单的讨论。

【外文】

1、

If we use the traditional notation $y = f ( x )$to indicate that the independent variable is $x$ and the dependent variable is $y$，then some common alternative notations for the derivative are as follows：

$f ^ { \prime } ( x ) = y ^ { \prime } = \frac { d y } { d x } = \frac { d f } { d x } = \frac { d } { d x } f ( x ) = D f ( x ) = D _ { x } f ( x )$

The symbols  $D $ and  $d/dx $ are called differentiation operators because they indicate the operation of differentiation，which is the process of calculating a derivative.

The symbol $d y / d x$，which was introduced by Leibniz，should not be regarded as a ratio

（for the time being）；it is simply a synonym for $\mathrm{f}(\mathrm{x})$.Nonetheless，it is a very useful and suggestive notation，especially when used in conjunction with increment notation.Refer-

ring to Equation 3.1.6（instantaneous rate of change$= \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x } = \lim _ { x _ { 2 } - x _ { 1 } } \frac { f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) } { x _ { 2 } - x _ { 1 } }$），we can rewrite the definition of derivative in Leibniz notation in the form $\frac { d y } { d x } = \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x }$

If we want to indicate the value of a derivative $d y / d x$ in Leibniz notation at a specific num-ber $a$，we use the notation $\left. \frac { d y } { d x } \right| _ { x = a } \quad$ or $\quad \frac { d y } { d x } ] _ { x - a }$

which is a synonym for$f ^ { \prime } ( a )$ .

参考文献：

[1]   James Stewart.Calculus(6th Edition)[M].Brooks Cole.Year,2007:126.

2、

To explain Leibniz's notation,we begin with a function $y=f(x)$ and write the difference quotient $\frac { f ( x + \Delta x ) - f ( x ) } { \Delta x }$

in the form

$\frac { \Delta y } { \Delta x }$

where $\Delta y = f ( x + \Delta x ) - f ( x )$.Here $\Delta y$ is not just any change in$y$；it is the specific change that results when the independent variable is changed from $x$ to $x + \Delta x$ .As we know，the difference quotient $\frac { \Delta y } { \Delta x }$ can be interpreted as the ratio of the change in $y$ to the change in $x$ along the curve $y = f ( x )$，and this is the slope of the secant（Fig，2.9）.Leibniz wrote the limit of this difference quotient，which of course is the derivative $f ^ { \prime } ( x )$，in the form $d y / d x$（read"$dy$ over$d x$"）.In this notation，the definition of the derivative becomes

 $\frac{d y}{d x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$  (1)

and this is the slope of the tangent in Fig.2.9.Two slightly different equivalent forms of  $dy/dx $ are  $\frac{d f(x)}{d x}$ and $\frac{d}{d x} f(x)$.

In the second of these，the symbol $d/dx$ should be thought of as an operation which can be applied to the function f(x) to yield its derivative   $f^{\prime}(x)$ ，as suggested by the equation$\frac{d}{d x} f(x)=f^{\prime}(x)$

The symbol $d/dx$ can be read"the derivative with respect to $x$ of..."whatever function of $x $ follows it.

It is important to understand that $dy/dx$ in (1) is a single indivisible symbol.In spite of the way it is written, it is not the quotient of two quantities $dy$ and $dx$, because $dy$ and $dx$ have not been defined and have no independent existence. In Leibniz's notation, the formation of the limit on the right of (1) is symbolically expressed by replacing the letter $\Delta$ by the letter$d$. From this point of view, the symbol $dy/dx$ for the derivative has the psychological advantage that it quickly reminds us of the whole process of forming the difference quotient $\Delta y / \Delta x$ and calculating its limit as $\Delta x \rightarrow 0$. There is also a practical advantage, for certain fundamental formulas developed in the next chapter are easier to remember and use when derivatives are written in the Leibniz notation.

参考文献：

[1]George Finlay.Simmons.Calculus with Analytic Geometry[M].The McGraw-Hill Companies.1996:60-61.

3、

The process of finding a derivative is called differentiation.You can think of differentiation as an operation on functions that associates a function $f^{\prime}$ with a function $f$.When the independent variable is $x$，the differentiation operation is also commonly denoted by

$f^{\prime}(x)=\frac{d}{d x}[f(x)]$ or $f^{\prime}(x)=D_{x}[f(x)]$

In the case where there is a dependent variable $\mathrm{y}=\mathrm{f}(\mathrm{x})$，the derivative is also commonly denoted by

$f^{\prime}(x)=y^{\prime}(x) \quad$ or $\quad f^{\prime}(x)=\frac{d y}{d x}$

$dx$ With the above notations，the value of the derivative at a point $x_{0}$ can be expressed as $f^{\prime}\left(x_{0}\right)=\left.\frac{d}{d x}[f(x)]\right|_{x=x_{0}}, \quad f^{\prime}\left(x_{0}\right)=\left.D_{x}[f(x)]\right|_{x=x_{0},}, f^{\prime}\left(x_{0}\right)=y^{\prime}\left(x_{0}\right), \quad f^{\prime}\left(x_{0}\right)=\left.\frac{d y}{d x}\right|_{x=x_{0}}$

Later, the symbols $dy$ and $dx$ will begiven specific meanings. However, for the time being do not regard $dy/dx$ asa ratio, but rather as a single symbol denoting the derivative.

参考文献：

[1]Howard Anton, Irl C. Bivens, Stephen Davis.Calculus(Tenth Edition).Wiley.2012:129.

4、

Definition. Let $d x$ denote an independent wariable which may take on any valuewhatsoever. Then the function of $x$ and $d x$ whose value is $f ^ { \prime } ( x ) d x$ is called the differential of $f$. Observe that the differential is a homogeneous linear function of $dx$; that is, for a fixed value of $x$, the differential has as its walue a fixed multiple of $dx$.

If we write $y=f(x)$, and if f is differentiable for a particular value of $x$, it is customary to write

$d y = f ^ { \prime } ( x ) d x$   (l)

so that dy is the value of the differential of $f$ for assigned values of $x$ and $dx$.

If we regard $x$ as fixed, $dy$ is a dependent variable whose value depends on the independent variable $dx$. The variables $dx$ and $dy$ are often refer-red to as the differentials of $x$ and $y$, respectively.The adjacent Fig.12 illustrates geometrically the functional dependence of $dy$ on $dx$, as well as the relation to the function f itself. The xyco-ordinate axes and the graph of the function $y=f(x)$ are shown in unbroken lines.A second co-ordinate system is showing with its origin at a typical point$( x , y )$ of the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$.The axes in this system are scales for the measurement of the variables $dx$，$dy$.The equation（1）has as its graph a straight line of slope $f ^ { \prime } ( x )$.This line is，of course，the tangent to the curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ at the origin of the $\mathrm{d} \mathrm{x}-\mathrm{d} \mathrm{y}$ co-ordinate system.

From（1）we have the quotient relation

$\frac { d y } { d x } = f ^ { \prime } ( x )$（2）

whenever $d x \neq 0$.The d-notations $dx$ and $dy$ go back to Leibniz's work in the seventeenth century，but Leibniz did not define the derivative by the limit of a quotient as we did in $f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$.It is to be emphasized that there is no need for $dx$ and $dy$ to be small in（2）.

参考文献：

[1]Taylor Angus & Wiley,Fayez.Advanced Calculus 3rd Edition[M].Library of Congress.2004：32-33.

5、

Definition Let $f(x)$ be a function defined in a neighborhood ${{N}_{r}}({{x}_{0}})$of a point ${{x}_{0}}$. Consider the ratio

$\phi (h)=\frac{f({{x}_{0}}+h)-f({{x}_{0}})}{h}$

Where h is a nonzero increment of ${{x}_{0}}$ such that -r<h<r .If $\phi (h)$ has a limit as h0,then this limit is called the derivative of f(x) at ${{x}_{0}}$ and is denoted by ${{f}^{'}}({{x}_{0}})$. It is also common to use the notation

${{\left. \frac{df(x)}{dx} \right|}_{x={{x}_{0}}}}=f'({{x}_{0}})$

[1]Khuri A.I.; Advanced Calculus With Applications In Statistics; Wiley Interscience:94

6、

If we use the raditional notation$y=f(x)$ to indicae that the independent variable is $x$and the dependent variable is $y$. then sone common altenative notations for the derivatve are as follows:

$f'(x)=y'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}f(x)=Df(x)={{D}_{x}}f(x)$

The symbols $D$and $\frac{d}{dx}$ are called dilfernthe Hon openatons because they indicale the openation of difereniatlon, which is the proces of calculating a derisative.

The symbol $\frac{dy}{dx}$, which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for $f'(x)$. Nonetheless, it is a very ueful and suggestive notation, espeially when used in conjunction with increment notation. Refrring to Equation 2.8.4, we can rewrite the definition of derivative in Leibniz notaion in the form

$\frac{dy}{dx}=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{\Delta y}{\Delta x}$

[1]James Stewart;Calculus(5th Edition);Brooks Cole.

7、

So far,given an open interval $I$ and a differentiable function $f:I\to R$,we have denoted the function’s derivative by

${F}':I\to R$,

So that ${{f}^{'}}(x)$is the derivative of $f:I\to R$ at $x$ in $I$.This notation has been completely adequate.However,as we introduce new classes of functions and when we study integration,certain formulas and algorithmic techniques become easier to assimilaye using analternate notation due to Leibnitz.Moreover ,this Leibnitz notation is widely used in texts on science and engineering ,so acquaintance with it is necessary.

For afifferentiable function $f:I\to R$,we denote ${{f}^{'}}(x)$ by

$\frac{dy}{dx}(f(x))$   or      $\frac{dy}{dx}$

[1]Patrick M. Fitzpatrick;Advanced Calculu(second edition);Brooks\/Cole:114.

8、

Although the meaning of these formulas is clear enough, attempts at literal interpretation are hindered by the reasonable stricture that an equation should not contain a function on one side and a number on the other. For example,if the third equation is to the true, theneither ${df\left( x \right)}/{dx}\;$ must denote ${{f}^{'}}(x)$,rather than${f}^{'}$,or else $2x$ must denote ,not anumber ,but the function whose value at $x$ is $2x$.It is really impossible to assert that one or the other of these laternatives is intended; in practice ${df\left( x \right)}/{dx}\;$ sometimes means${f}^{'}$and sometimes means${{f}^{'}}(x)$,while $2x$ may denote either a number or a function .Because of this ambiguity, most authors are reluctant to denote ${{f}^{'}}(a)$ by

$\frac{df(x)}{dx}(a)$

Instead${{f}^{'}}(a)$ is usually denoted by the barbaric,but unambiguous, symbol

$\frac{df(x)}{dx}{{|}_{x=a}}$

[1]   Michael Spivak;Calculus;Publish or Perish:141

9、

函数$f:R\to R$在$a\in R$点可微，如果有一线性变换$\lambda :R\to R$使得

$\underset{h\to 0}{\mathop{\lim }}\,\frac{f(a+h)-f(a)-\lambda (h)}{h}=0$

参考文献：斯皮瓦克. 流形上的微积分(齐民友译)[M]. 人民邮电出版社，2006:16

10、

 定义5.1 设$f$在开区间$\left( a,b \right)$上定义，并假定$c\in \left( a,b \right)$，则$f$称为在$c$点是可微的，只要极限

$\underset{x\to c}{\mathop{\lim }}\,\frac{f(x)-f(c)}{x-c}$

存在，这个极限用${{f}^{'}}(c)$表示，称为$f$在$c$点的导数。

参考文献：

[1] TOM M.Appostol. 数学分析（原书第二版）（刑富冲译）[M]. 机械工业出版社， 2006:104