Differentiation Techniques and Applications | 微分技巧与应用

📚 Differentiation Techniques and Applications | 微分技巧与应用

Differentiation is one of the central pillars of Edexcel A-Level Mathematics. It allows us to measure how a function changes at any instant and is essential for curve sketching, optimisation, kinematics and modelling. This revision guide walks through the key rules, standard derivatives and common exam applications.

微分是爱德思 A-Level 数学的核心支柱之一。它让我们能够度量函数在任意瞬间的变化情况，对于曲线作图、优化问题、运动学以及数学建模都至关重要。本复习指南将系统梳理关键求导法则、标准导数公式以及常见考试应用。

1. The Gradient of a Curve and First Principles | 曲线的斜率与第一原理

The derivative of a function f at a point x is defined as the limit of the average gradient between two nearby points as the distance between them tends to zero.

函数 f 在点 x 处的导数定义为两点间平均斜率当两点距离趋于零时的极限。

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

In Edexcel papers, you may be asked to evaluate this limit for a simple polynomial such as f(x) = x². Expanding and simplifying before letting h → 0 is the standard approach.

在爱德思考试中，可能会要求你对简单多项式（如 f(x) = x²）计算这一极限。标准方法是先展开并化简，再令 h → 0。

This formal definition underpins all differentiation rules. Even if exam questions usually focus on applying rules, understanding first principles helps justify the methods and handle unfamiliar functions.

这一形式定义是所有求导法则的基础。尽管考试题目通常侧重于应用法则，理解第一原理有助于证明方法的合理性并处理陌生函数。

2. Basic Differentiation Rules | 基本求导法则

For any real constant n, the power rule states:

对于任意实数常量 n，幂函数求导法则为：

If y = xⁿ, then dy/dx = n xⁿ⁻¹.

The constant multiple rule and sum rule allow us to differentiate linear combinations term by term.

常数倍法则和加法法则允许我们逐项微分线性组合。

For example, if y = 5x³ − 2x +

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