Edexcel A-Level Pure Mathematics: Differentiation and Integration Masterclass | Edexcel A-Level 纯数学：微分与积分核心突破

📚 Edexcel A-Level Pure Mathematics: Differentiation and Integration Masterclass | Edexcel A-Level 纯数学：微分与积分核心突破

This revision note covers the core techniques of differentiation and integration required for the Edexcel A-Level Mathematics specification. Understanding these methods is essential for success in Pure Mathematics papers and for applied contexts such as kinematics and optimisation.

本文复习笔记涵盖 Edexcel A-Level 数学大纲要求的微分与积分核心技巧。掌握这些方法对于纯数学试卷以及运动学、最优化等应用情境至关重要。

1. The Chain Rule | 链式法则

The chain rule differentiates composite functions of the form y = f(g(x)). It states that dy/dx = dy/du × du/dx, where u = g(x). This rule is used extensively in A-Level questions involving powers, exponentials, logarithms and trigonometric functions.

链式法则用于对 y = f(g(x)) 形式的复合函数求导。它指出 dy/dx = dy/du × du/dx，其中 u = g(x)。该法则广泛用于涉及幂函数、指数函数、对数函数和三角函数的 A-Level 考题。

dy/dx = dy/du × du/dx

Example: If y = (2x³ − 5)⁶, set u = 2x³ − 5. Then dy/du = 6u⁵ and du/dx = 6x², so dy/dx = 36x²(2x³ − 5)⁵.

示例：若 y = (2x³ − 5)⁶，令 u = 2x³ − 5，则 dy/du = 6u⁵，du/dx = 6x²，因此 dy/dx = 36x²(2x³ − 5)⁵。

2. Product and Quotient Rules | 乘法法则与除法法则

The product rule is used when differentiating a function that is the product of two simpler functions, u(x)v(x). The quotient rule applies when one function is divided by another.

乘法法则用于对两个简单函数的乘积 u(x)v(x) 求导。除法法则适用于一个函数除以另一个函数的情形。

Product rule: d/dx (uv) = u dv/dx + v du/dx
Quotient rule: d/dx (u/v) = (v du/dx − u dv/dx) / v²

Example: For y = x²eˣ, take u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, so dy/dx = x²eˣ + 2xeˣ = eˣ(x² + 2x).

示例：对于 y = x²eˣ，取 u = x²、v = eˣ，则 u’ = 2x、v’ = eˣ，因此 dy/dx = x²eˣ + 2xeˣ = eˣ(x² + 2x)。

3. Implicit Differentiation | 隐函数求导

When y cannot be easily written as an explicit function of x, differentiate both sides of the equation with respect to x and apply the chain rule to any term involving y.

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