新导数定义下的、不涉及分母的无矛盾最简求导法

The Noncontradictory Simplest Derivative Method under the Definition of A New Derivative that Does Not Involve the Denominator

摘要: 通过前期笔者的系列文章，彻底搞清了牛顿等当年究竟是如何求出正确导数的，尽管表面看来它会导致贝克莱悖论。当然，这需要一个新的导数定义，尽管牛顿等实际求出的就是也只能是它。在此新定义的基础上，提出与之一致的没有分母的求导新法，这是迄今为止最简单的、同时无矛盾的求导方法。在讨论中发现，很多论者惑于增量比值函数分母上的那个自变量的实质，尽管笔者已经在逻辑上彻底阐明了。于是，这里根据线性或其增量方程的系数就是其斜率的众所周知的数学事实，直接针对并无自变量分母的增量函数而不再是有自变量分母的增量比值函数来求导，也就是只要求得其系数即可知其斜率。于是自然可以彻底告别分母上的自变量，彻底杜绝任何歧义性的可能：没有那个自变量分母，照样求出了导数。勿再纠结于分母上的那个自变量了，它在推导的一开始就不存在。同时，在新的导数定义下，我们可以得到与之协调的微分新定义，同样消除了函数与自变量微分定义上的不协调，进而解决了积分中的隐含矛盾。最后，笔者进一步揭示了极限法求导(所谓“第二代微积分”)隐含的逻辑矛盾，因此根本不成立，贝克莱悖论不可能由极限法消除。因此，第一代微积分(牛顿等)实际是对的，应该为之正名。

Abstract: Through the previous series of articles of the author, it is completely clear how Newton and others were able to find the correct derivative, although it appears that it will lead to Berkeley paradox. And, of course, that requires a new definition of the derivative, although Newton et al actually figured that out. On the basis of this new definition, a new derivative method without denominator is proposed, which is the simplest and non-contradictory derivative method so far. At the same time, under the new definition of derivative, we can get a new definition of derivative that is compatible with it, which also eliminates the inconsistency between the definition of derivative of function and independent variable, thus solving the implicit contradiction in integration. Finally, the author further reveals the logic contradiction implied by the derivation of limit method (the so-called “second generation calculus”), so it is not valid at all, and Berkeley paradox cannot be eliminated by limit method. So the first calculus (Newton et al.) was actually right and should be vindicated.