Edexcel A Level Maths: Differentiation Rules, Applications and Exam Techniques | 爱德思A Level数学：微分法则、应用与应试技巧

📚 Edexcel A Level Maths: Differentiation Rules, Applications and Exam Techniques | 爱德思A Level数学：微分法则、应用与应试技巧

Differentiation is one of the most heavily examined topics in Edexcel A Level Mathematics. It appears in Pure Mathematics papers, supports mechanics problems such as velocity and acceleration, and underpins many real-world optimisation questions. This revision guide brings together the key rules, worked examples, and exam-focused advice you need to master differentiation from first principles to parametric equations.

微分是爱德思 A Level 数学中考查最频繁的主题之一。它出现在纯数学试卷中，支撑力学中的速度与加速度问题，也是许多现实优化题的数学基础。本篇复习指南汇集了从第一性原理到参数方程微分的核心法则、典型示例与应试建议。

1. The Gradient Function and First Principles | 梯度函数与第一性原理

The derivative f'(x) is the gradient function of a curve y = f(x). It gives the instantaneous rate of change of y with respect to x at any point where the function is smooth.

导数 f'(x) 是曲线 y = f(x) 的梯度函数。它给出了函数光滑处任意点上 y 关于 x 的瞬时变化率。

The formal definition from first principles is:

第一性原理的形式化定义如下：

f'(x) = lim_h→0 [f(x + h) – f(x)] / h

Edexcel often asks candidates to use this limit definition to differentiate simple functions such as x², x³ or 1/x. Expanding and simplifying before letting h tend to zero is essential.

爱德思考试常要求考生使用该极限定义对 x²、x³ 或 1/x 等简单函数求导。在令 h 趋近于 0 之前，必须先展开并化简。

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 approaches zero, the expression tends to 2x.

例如，若 f(x) = x²，则 f(x + h) – f(x) = (x + h)² – x² = 2xh + h²。除以 h 后得到 2x + h，当 h 趋近于 0 时，该表达式趋近于 2x。

2. Standard Derivatives and the Power Rule | 标准导数与幂法则

For any real power n, the power rule states that if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. This rule is the foundation for polynomials and many rational functions.

对任意实数次幂 n，幂法则指出：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。该法则是多项式及许多有理函数求导的基础。

Key standard derivatives are summarised below:

主要标准导数总结如下：

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