Edexcel A-Level Maths: Mastering Differentiation | 爱德思 A-Level 数学：掌握微分法

📚 Edexcel A-Level Maths: Mastering Differentiation | 爱德思 A-Level 数学：掌握微分法

Differentiation is one of the most frequently examined topics in Edexcel A-Level Pure Mathematics. It provides the tools to measure instantaneous rates of change, determine gradients of curves, and solve real-world optimisation problems. A solid command of derivative rules, stationary points, and tangents is essential for high marks across Paper 1 and Paper 2.

微分是爱德思 A-Level 纯数学中最常考的专题之一。它提供了度量瞬时变化率、求曲线斜率和解决现实优化问题的工具。扎实掌握求导法则、平稳点与切线，是在 Paper 1 和 Paper 2 中取得高分的关键。

1. First Principles and the Definition of Derivative | 导数定义与第一原理

The derivative is defined as the limit of the average rate of change over a shrinking interval. For a function f(x), the derivative at a point is written f'(x) and is given by the following limit, provided it exists:

f'(x) = lim(h→0) [f(x+h) – f(x)] ÷ h

导数定义为区间不断缩小时平均变化率的极限。对于函数 f(x)，某点处的导数记为 f'(x)，在极限存在的条件下由以下表达式给出：

f'(x) = lim(h→0) [f(x+h) – f(x)] ÷ h

To differentiate from first principles, substitute f(x+h), expand, cancel h, and then let h tend to zero. For example, if f(x)=x², then f(x+h)-f(x) = (x+h)² – x² = 2xh + h², so the limit becomes 2x.

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