Mastering Differentiation for Edexcel A-Level Pure Maths | 精通 Edexcel A-Level 纯数微分

📚 Mastering Differentiation for Edexcel A-Level Pure Maths | 精通 Edexcel A-Level 纯数微分

Differentiation is one of the most heavily examined topics across Edexcel A-Level Mathematics. It underpins tangent equations, stationary-point analysis, optimisation, kinematic modelling and connected rates of change. This tutorial consolidates the techniques and exam language you need to solve Pure Mathematics questions confidently.

微分是 Edexcel A-Level 数学中考查频率最高的主题之一。它是切线方程、驻点分析、最优化、运动学建模以及相关变化率问题的基础。本教程将帮助你巩固相关技巧和考试表述，使你能够自信地解答纯数试题。

1. First Principles and the Derivative as a Limit | 第一性原理与作为极限的导数

At AS Level, Edexcel often asks you to differentiate a simple function from first principles using the limit definition of the gradient of a chord. The derivative is defined by

在 AS 阶段，Edexcel 经常要求考生用弦的斜率极限定义，从第一性原理出发求一个简单函数的导数。导数定义为

f'(x) = lim(h→0) [f(x + h) − f(x)] / h

You must show the full expansion, simplify the numerator, cancel h, and then let h tend to 0. Do not simply write the standard result.

你必须展示完整展开、化简分子、约去 h，再令 h 趋向于 0，而不是直接写出标准结果。

For example, if f(x) = x², expand (x + h)² = x² + 2xh + h², subtract x², divide by h, and obtain 2x + h. As h → 0, the limit is 2x.

例如，若 f(x) = x²，展开 (x + h)² = x² + 2xh + h²，减去 x²，除以 h，得到 2x + h。当 h → 0 时，极限为 2x。

2. Standard Results and the Power Rule | 标准结果与幂函数法则

Edexcel assumes fluency with standard derivatives of powers, exponentials, logarithms and trigonometric functions. The power rule states that multiplying by the old power and reducing the power by one gives the derivative of xⁿ.

Edexcel 要求考生熟练掌握幂函数、指数函数、对数函数和三角函数的标准化导数。幂函数法则指出，将原幂次乘到前面，再把幂次减一，即可得到 xⁿ 的导数。

Remember that constants differentiate to zero, and terms can be differentiated

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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