📚 The Results of the War | 微积分战争的结果
In the late 17th century, two great minds, Isaac Newton and Gottfried Wilhelm Leibniz, independently developed the foundations of calculus. Their bitter priority dispute, often called the ‘Calculus War’, shaped the way we learn and use calculus today. For A-Level Edexcel Mathematics students, the results of this historical war are not just academic gossip: they appear in the notation, rules, and applications you meet in differentiation and integration.
17 世纪末，两位伟人——艾萨克·牛顿和戈特弗里德·威廉·莱布尼茨——各自独立创立了微积分的基础。他们之间激烈的优先权之争，常被称为“微积分战争”，塑造了我们今天学习和使用微积分的方式。对于 A-Level Edexcel 数学的学生来说，这场历史战争的结果不仅仅是学术八卦：它们体现在你在微分和积分中遇到的符号、法则和应用之中。
1. Background of the Calculus War | 微积分战争的背景
Newton began developing his ‘method of fluxions’ in the 1660s while at Cambridge, but he published very little about it for many years. Leibniz started working on similar ideas in the 1670s in Paris and published his first paper on differential calculus in 1684.
牛顿在 17 世纪 60 年代于剑桥开始发展他的“流数术”，但多年来很少发表相关内容。莱布尼茨在 17 世纪 70 年代于巴黎开始研究类似的思想，并于 1684 年发表了关于微分学的第一篇论文。
Because Leibniz published first, many continental European mathematicians adopted his notation and methods rapidly. Newton’s supporters later accused Leibniz of plagiarism, leading to a prolonged and hostile dispute between British and continental mathematicians.
由于莱布尼茨率先发表，许多欧洲大陆数学家迅速采用了他的符号和方法。牛顿的支持者后来指控莱布尼茨抄袭，导致了英国与欧洲大陆数学家之间旷日持久的敌对争论。
This dispute had practical consequences: British mathematics became isolated for over a century, while continental mathematicians advanced faster using Leibniz’s notation.
这场争论产生了实际后果：英国数学在一个多世纪里变得孤立，而欧洲大陆数学家使用莱布尼茨的符号取得了更快的进步。
2. Newton’s Approach: Fluxions | 牛顿的方法：流数术
Newton thought of curves as being generated by the motion of a point. He called a changing quantity a ‘fluent’ and its rate of change a ‘fluxion’. For a fluent y, he denoted its fluxion by ẏ (y with a dot above).
牛顿将曲线看作由点的运动生成。他把变化的量称为“流数”（fluent），其变化率称为“流数”（fluxion）。对于一个流数 y，他用 ẏ（y 上方加一点）表示其流数。
Newton’s notation is still used in physics today, especially for derivatives with respect to time. For example, if s is displacement, then ṡ is velocity and s̈ is acceleration.
牛顿的符号至今仍在物理学中使用，尤其是对时间的导数。例如，若 s 是位移，则 ṡ 是速度，s̈ 是加速度。
However, Newton’s dot notation is less convenient when dealing with functions of a general variable x, because it does not explicitly show the variable of differentiation.
然而，牛顿的点符号在处理一般变量 x 的函数时不太方便，因为它没有明确显示微分的变量。
3. Leibniz’s Notation: dy/dx | 莱布尼茨的符号：dy/dx
Leibniz introduced the notation dy/dx for the derivative of y with respect to x. He viewed dy and dx as infinitesimally small changes in y and x, and their ratio as the slope of the tangent.
莱布尼茨引入了 dy/dx 表示 y 关于 x 的导数。他把 dy 和 dx 看作 y 和 x 的无穷小变化，它们的比值就是切线的斜率。
This notation is highly practical for the chain rule: dy/dx = dy/du × du/dx. The symbols behave like fractions, making the rule easy to remember and apply.
这种符号对链式法则非常实用：dy/dx = dy/du × du/dx。符号像分数一样运算，使得规则易于记忆和应用。
Edexcel A-Level Mathematics uses Leibniz’s notation throughout the differentiation and integration topics. You will see dy/dx for first derivative, d²y/dx² for second derivative, and ∫ f(x) dx for integrals.
Edexcel A-Level 数学在微分和积分主题中全程使用莱布尼茨符号。你会看到 dy/dx 表示一阶导数，d²y/dx² 表示二阶导数，∫ f(x) dx 表示积分。
4. The Priority Dispute | 优先权之争
The dispute over who invented calculus first began in earnest in 1699 and intensified after 1704. Newton’s followers claimed Leibniz had seen Newton’s unpublished work during a visit to London.
关于谁首先发明了微积分的争论于 1699 年正式开始，并在 1704 年后加剧。牛顿的追随者声称莱布尼茨在访问伦敦期间见过牛顿未发表的手稿。
In 1712, the Royal Society, under Newton’s presidency, published a report supporting Newton’s priority. This was not an impartial judgement, but it damaged Leibniz’s reputation in Britain.
1712 年，英国皇家学会在牛顿担任会长期间发表了一份支持牛顿优先权的报告。这不是一份公正的裁决，但它损害了莱布尼茨在英国的名誉。
Modern historians generally agree that Newton and Leibniz developed calculus independently. Newton’s work was earlier, but Leibniz’s publication was earlier and his notation proved more influential.
现代历史学家普遍认为牛顿和莱布尼茨是独立发展出微积分的。牛顿的工作更早，但莱布尼茨的发表更早，而且他的符号最终更具影响力。
5. Result 1: Standard Notation | 结果一：标准符号体系
One lasting result of the war is that modern calculus uses Leibniz’s notation as the international standard. In your Edexcel exam, derivatives are written as dy/dx, f'(x), or d/dx ( … ), and integrals as ∫ … dx.
这场战争的一个持久结果是现代微积分使用莱布尼茨符号作为国际标准。在你的 Edexcel 考试中，导数写作 dy/dx、f'(x) 或 d/dx ( … )，积分写作 ∫ … dx。
For example, if y = x³ + 2x, then dy/dx = 3x² + 2. The notation shows clearly that y is being differentiated with respect to x.
例如，若 y = x³ + 2x，则 dy/dx = 3x² + 2。该符号清楚地表明 y 正在对 x 求导。
- First derivative: f'(x) or dy/dx
- Second derivative: f”(x) or d²y/dx²
- Definite integral: ∫ₐᵇ f(x) dx
一阶导数：f'(x) 或 dy/dx；二阶导数：f”(x) 或 d²y/dx²；定积分：∫ₐᵇ f(x) dx。
6. Result 2: Differentiation Rules | 结果二：微分法则
The standard rules of differentiation that you use in A-Level Mathematics are direct descendants of Leibniz’s approach. The power rule, product rule, quotient rule, and chain rule are all expressed naturally using dy/dx notation.
你在 A-Level 数学中使用的标准微分法则都是莱布尼茨方法的直接产物。幂法则、乘积法则、商法则和链式法则都能用 dy/dx 符号自然表达。
Power rule: d/dx (xⁿ) = n xⁿ⁻¹
幂法则：d/dx (xⁿ) = n xⁿ⁻¹
Product rule: d/dx (uv) = u’v + uv’
乘积法则：d/dx (uv) = u’v + uv’
Quotient rule: d/dx (u/v) = (u’v – uv’) / v²
商法则：d/dx (u/v) = (u’v – uv’) / v²
Chain rule: dy/dx = dy/du × du/dx
链式法则：dy/dx = dy/du × du/dx
These rules are essential for Edexcel Pure Mathematics and are tested in questions on differentiation from first principles, stationary points, and optimisation.
这些法则对 Edexcel 纯数学至关重要，在从第一原理求导、驻点和优化等问题中都会考查。
7. Result 3: The Fundamental Theorem | 结果三：微积分基本定理
The fundamental theorem of calculus links differentiation and integration as inverse processes. This key idea was recognised by both Newton and Leibniz, but Leibniz’s notation makes the connection transparent.
微积分基本定理将微分和积分联系为互逆的过程。牛顿和莱布尼茨都认识到了这一关键思想，但莱布尼茨的符号使这种联系清晰可见。
If F(x) is an antiderivative of f(x), then:
∫ₐᵇ f(x
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