Mastering Differentiation for Edexcel A-Level Pure Maths | 精通微分：Edexcel A-Level 纯数学

📚 Mastering Differentiation for Edexcel A-Level Pure Maths | 精通微分：Edexcel A-Level 纯数学

Differentiation is one of the most examined topics in Edexcel A-Level Pure Mathematics. It underpins ideas of rates of change, gradients of curves, optimisation and modelling. This revision guide covers the essential definitions, rules, applications and common pitfalls you must master for Papers 1 and 2.

微分是 Edexcel A-Level 纯数学中考查频率最高的主题之一。它是变化率、曲线斜率、最优化和建模等概念的基础。本复习指南涵盖你必须掌握的核心定义、法则、应用以及常见失分点，帮助你应对 Paper 1 和 Paper 2。

1. Definition from First Principles | 第一原理定义

In Edexcel Pure Maths, the derivative of a function f(x) is defined as the limit of the average gradient between two points on the curve as the gap h tends to zero. This formal limit is the reason differentiation gives an instantaneous rate of change.

在 Edexcel 纯数学中，函数 f(x) 的导数定义为曲线上两点之间的平均斜率在间距 h 趋于 0 时的极限。这个形式化极限正是微分能给出瞬时变化率的原因。

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

For example, if f(x) = x², then f(x+h) − f(x) = (x+h)² − x² = 2xh + h². Dividing by h gives 2x + h, and as h → 0 the limit is 2x, so f'(x) = 2x.

例如，若 f(x) = x²，则 f(x+h) − f(x) = (x+h)² − x² = 2xh + h²。除以 h 得到 2x + h，当 h → 0 时极限为 2x，因此 f'(x) = 2x。

2. Notation and Key Terminology | 符号与关键术语

Edexcel exam papers often switch between Lagrange notation f'(x) and Leibniz notation dy/dx. You must be able to read both and know that d/dx is an operator that means differentiate with respect to x.

Edexcel 试卷中常交替使用拉格朗日记号 f'(x) 和莱布尼茨记号 dy/dx。你必须能读懂两者，并知道 d/dx 是一个算子，表示对 x 求导。

If y = f(x), then the derivative can be written as y’, f'(x), dy/dx, or d/dx [f(x)]. The second derivative is written f”(x) or d²y/dx².

若 y = f(x)，则导数可以写成 y’、f'(x)、dy/dx 或 d/dx [f(x)]。二阶导数写作 f”(x) 或 d²y/dx²。

3. Differentiating Powers of x | x 的幂的微分

The most frequently used rule in A-Level differentiation is the power rule. For any real constant n, if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.

A-Level 微分中最常用的法则是幂法则。对于任何实数常量 n，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。

d/dx (xⁿ) = n xⁿ⁻¹

This rule works for positive integers, negative powers and fractional powers. For example, x⁵ differentiates to 5x⁴, x⁻³ differentiates to −3x⁻⁴, and x^(1/2

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

更多咨询请联系16621398022（同微信）