Calculus Core Revision Guide for A-Level Mathematics | A-Level 数学备考：微积分核心重点解析

📚 Calculus Core Revision Guide for A-Level Mathematics | A-Level 数学备考：微积分核心重点解析

Calculus is the branch of mathematics that studies continuous change. In A-Level Mathematics, differentiation and integration appear in almost every exam paper, and mastering their core rules is essential for a high grade. This guide condenses the key techniques you need into twelve focused sections.

微积分是研究连续变化的数学分支。在 A-Level 数学中，微分与积分几乎出现在每份试卷中，掌握其核心规则是获得高分的关键。本指南将你需要掌握的关键技巧浓缩为十二个专题小节。

1. Differentiation from First Principles and the Power Rule | 第一性原理求导与幂法则

Differentiation measures the rate at which a function changes. From first principles, the derivative f'(x) is defined as the limit of the average rate of change as h tends to zero.

微分衡量函数变化的速率。根据第一性原理，导数 f'(x) 定义为当 h 趋近于零时平均变化率的极限。

f'(x) = lim[h→0] (f(x+h) − f(x)) / h

The power rule states that if y = xⁿ, then dy/dx = nxⁿ⁻¹. This rule also works for negative and fractional powers, which makes it one of the most frequently used rules in A-Level papers.

幂法则指出：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。该法则对负指数和分数指数同样适用，因此是 A-Level 试卷中最常使用的法则之一。

For example, if y = x³, then dy/dx = 3x². If y = √x = x½, then dy/dx = ½x⁻½.

例如，若 y = x³，则 dy/dx = 3x²。若 y = √x = x½，则 dy/dx = ½x⁻½。

2. Product, Quotient and Chain Rule | 乘积法则、商法则与链式法则

The product rule is used when two functions are multiplied. If y = uv, then dy/dx = u·dv/dx + v·du/dx.

乘积法则用于两个函数相乘的情形。若 y = uv，则 dy/dx = u·dv/dx + v·du/dx。

The quotient rule is used for fractions. If y = u/v, then dy/dx = (v·du/dx − u·dv/dx) / v².

商法则用于分式情形。若 y = u/v，则 dy/dx = (v·du/dx − u·dv

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