Mixed Exercise 9: Differentiation Masterclass | 混合练习9：微分专题精讲

📚 Mixed Exercise 9: Differentiation Masterclass | 混合练习9：微分专题精讲

Chapter 9 of Edexcel Pure Mathematics Year 2 consolidates the differentiation of trigonometric, exponential and logarithmic functions, and introduces the chain rule, product rule and quotient rule. This article mirrors the structure of Mixed Exercise 9, presenting revision notes and worked-style examples that target the most commonly tested skills in the A-Level examination.

Edexcel 纯数学 Year 2 第9章系统讲解了三角、指数与对数函数的微分法则，并引入了链式法则、乘积法则与商法则。本文紧随《混合练习9》的题型结构，为你梳理高频考点、核心方法与易错警示，直击 A-Level 考试中最常见的考查方式。

1. Core Standard Derivatives | 基本导数公式回顾

Before tackling mixed questions, recall the four standard results. For y = sin x, the derivative is cos x; for y = cos x, the derivative is -sin x; for y = eˣ, the derivative is eˣ; and for y = ln x, the derivative is 1/x. These results must be memorised exactly, with signs and domains correct.

在进入混合练习之前，先回顾四个基本求导结果。若 y = sin x，则导数为 cos x；若 y = cos x，则导数为 -sin x；若 y = eˣ，则导数为 eˣ；若 y = ln x，则导数为 1/x。这些公式必须精确记忆，尤其是符号与定义域不能出错。

d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x

Using the chain rule, these extend to composite functions: d/dx[sin(kx)] = k cos(kx), d/dx[cos(kx)] = -k sin(kx), d/dx[e^(kx)] = k e^(kx), and d/dx[ln(kx)] = 1/x. Note that the last result is independent of the constant k, which surprises many students in mock examinations.

利用链式法则，这些公式可以推广到复合函数：d/dx[sin(kx)] = k cos(kx)，d/dx[cos(kx)] = -k sin(kx)，d/dx[e^(kx)] = k e^(kx)，而 d/dx[ln(kx)] = 1/x。注意最后一个结果与常数 k 无关，这一点常令许多学生在模考中感到意外。

2. Differentiating Trigonometric Functions | 三角函数微分

Exam questions

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