00后大学生：函数的导数不用极限概念来定义

        在互联网大课堂上，你们的Keisler老师说：函数导数不用极限概念来定义。他给出以下定义：

        DIFFERENTIATION（第二章微分学）

2.1 DERIVATIVES（导数）

We are now readyto explain what is meant by the slope of a curve or the velocity of a movingpoint. Consider a real function f and a real number a in the domain of f When xhas value a,f(x) has value f(a). Now suppose the value of x is changed from ato a hyperreal number a + Δx which isinfinitely close to but not equal to a. Then the new value of f(x) will be f(a+ Δx). In this process thevalue of x will be changed by a nonzero infinitesimal amount Δx, while the value of f(x) will be changed by the amount

        f(a + Δx) - f(a).

The ratio of thechange in the value of f(x) to the change in the value of x is

    （f(a + Δx) - f(a)）/ Δx  

This ratio isused in the definition of the slope of f which we now give.

    DEFINITION

S is said to bethe slope of f at a if S = st (f(a + Δx) - f(a))/ ΔX)

        for every nonzero irifinitesimal Δx. The slope, whenit exists, is infinitely close to the ratio of the change in f(x) to aninfinitely small change in x. Given a curve y = f(x), the slope of f at a isalso called the slope of the curve y =f（x）。

        注意：上述函数符号st（x）是提取超实数x的标准部分的意思，在第一章最后一节有定义。

实际上，函数曲线在一点的斜率就是该函数在这一点的导数。在无穷小微积分中，函数的导数不需要使用极限概念来定义。这个认识必须不断灌输。

        注意：幸运的是，这个现代版的莱布尼兹关于导数的定义就在你的指尖上，它存储在定期备份的加密互联网云端。有人企图删除它，这是白日做梦。

        为此，00后大学生向Keisler老师致敬！Keisler教授指导你们赢在起跑线上！

袁萌   6月18日