【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。

【美国】

1、

微分

有许多方式来记函数$y=f(x)$的导数，除了$f'(x)$外，最常用的记号有：

$y'$念作“$y$撇”            很好且简洁的表示，但没有说出自变量是什么

$\frac{dy}{dx}$念作“$dy$”“$dx$”      说出了自变量并用$d$来表示导数

$\frac{df}{dx}$念作“$df$”“$dx$”      强调了函数的名称

$\frac{d}{dx}f(x)$念作“$ddx$”“$f(x)$”强调了微商是作用在$f$上的一种运算概念

我们也把$\frac{dy}{dx}$念作“$y$关于$x$的导数”而把$\frac{df}{dx}$和$\frac{d}{dx}f(x)$念作“$f$关于$x$的导数”

[1] Thomas;托马斯微积分(第10版);叶其孝,王耀东,唐兢译；高等教育出版社:165.

2、

 We call this limit of the difference quotient the derivative of the function $y=f(x)$ at the point $x$. We shall generally use either the notation of Lagrange, $y'=f'(x)$，to denote the derivative, or, as Leibnitz did, the symbol$\frac{dy}{dx}$or $\frac{df(x)}{dx}$ or $(\frac{d}{dx})f(x)$

In Leibnitz's notation the passage to the limit in the process of differentiation is symbolically expressed by replacing the symbol$\Delta $ by the symbol $d$, motivating Leibnitz's symbol for the derivative defined by the equation

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

If we wish to obtain a clear grasp of the meaning of the differential calculus, we must beware of the old fallacy of imagining the derivative as the quotient of two“quantities" $dy$ and $dx$ which are actually “infinitely small." The diference quotient $\frac{\Delta y}{\Delta x}$ has a meaning only for differences $\Delta x$ which are not equal to zero. After forming this genuine difference quotient we must perform the passage to the limit by means of a transformation or some other device which also in the limit avoids division by zero. It does not make sense to suppose that first $\Delta x$ and $\Delta y$ go through something like a limiting process and reach values which are infinitesimally small but still not zero, so that $\Delta x$ and $\Delta y$ are replaced by "infinitely small quantities" or“infinitesimals" $dx$ and $dy$, and that the quotient of these quantities is then formed. Such a conception of the derivative is incompatible with mathematical clarity; in fact, it is entirely meaningless. For many people it undoubtedly has a certain charm of mystery, always associated with the word“infinite"; in the early days of the differential calculus even Leibnitz himself was capable of combining these vague mystical ideas with a thoroughly clear handling of the limiting process. But today the mysticism of infinitely small quantities has no place in the calculus.

For any differentiable function $f$ and for a fixed $x$ this differential is a well-defined linear function of h=$\Delta $x. For example

$\lim _{\Delta x \rightarrow 0} \frac{\Delta f(x)}{\Delta x}=c$

                                         $c \neq 0$                                      

                                  $\frac{\Delta f}{d f} \rightarrow 1$                     

                                        $\Delta x \rightarrow 0$                                    

                                                 $\gamma(\Delta x)$                                          

                                      $\Delta f(x)$                                      

                                            $\delta$                                        

                                    $\gamma(0)=0$                                  

                                     $\Delta x=d x$                                    

for the function $\mathrm{y}=\mathrm{x}^{2}$ we have $\mathrm{d} \mathrm{y}=\mathrm{d}\left(x^{2}\right)=2 x \Delta x=2 x h$. For the particular function  $y=x$  whose  derivative   has  the  constant  value  one, we simply have $\mathrm{d} \mathrm{x}=\Delta x$. It is then consistent with our definition to write $\mathrm{d} \mathrm{x}$  for $\Delta x$ when $x$ is the independent variable; hence the differential of any function $\mathrm{y}=\mathrm{f}(x)$ can also be written as    

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

The increment of the dependent variable

$\Delta y=f^{\prime}(x) d x+\varepsilon d x=\mathrm{d} y+\varepsilon d x$

Differs from the differential $\mathrm{d} y$ by the amount $\varepsilon dx$,which in general is not zero. In the example  of the function $\mathrm{y}=x^{2}$,we have $\mathrm{d} \mathrm{y}=2 \mathrm{xdx}$,whereas

$\Delta y={{(x+dx)}^{2}}-{{x}^{2}}=2xdx+{{(dx)}^{2}}=dy+\varepsilon dx$

Where  $\varepsilon =dx$

[1] Richard Courant,Fritz John; Introduction to Calculus and Analysis[M];Interscience Publishers:171-180.

3、

   如果${f}'$在$x$点有定义，便说$f$在$x$点可微（或可导），如果${f}'$在集$E\subset \left[ a,b \right]$的每一个点有定义，便说$f$在$E$上可微。

[1]Rudin;数学分析原理第一册(原书第三版)[M];赵慈庚，蒋铎译；人民教育出版社：115.