Mastering Differentiation and Integration for Edexcel A-Level Pure Maths | 精通微分与积分：Edexcel A-Level 纯数学

📚 Mastering Differentiation and Integration for Edexcel A-Level Pure Maths | 精通微分与积分：Edexcel A-Level 纯数学

Calculus is one of the most heavily weighted areas in Edexcel A-Level Pure Mathematics. It connects algebra, trigonometry and graphs, and appears across Paper 1 and Paper 2. This article reviews the key rules, applications and exam strategies for differentiation and integration.

微积分是 Edexcel A-Level 纯数学中分值最高的模块之一。它连接代数、三角函数和图像，出现在 Paper 1 和 Paper 2 中。本文将复习微分与积分的核心法则、应用和考试策略。

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

The derivative of a function f(x) is defined as the limit of the average rate of change over an interval as the interval width tends to zero. This is known as differentiation from first principles.

函数 f(x) 的导数定义为区间宽度趋近于零时平均变化率的极限。这就是”第一原理求导”。

f′(x) = limₕ→₀ (f(x+h) – f(x)) / h

For a simple quadratic f(x) = x², the numerator becomes (x+h)² – x² = 2xh + h². Dividing by h gives 2x + h, and the limit as h tends to 0 is 2x. Therefore f′(x) = 2x.

对于简单二次函数 f(x) = x²，分子变为 (x+h)² – x² = 2xh + h²。除以 h 得到 2x + h，当 h 趋于 0 时极限为 2x。因此 f′(x) = 2x。

In Edexcel exams, first principles questions usually ask you to prove a standard result such as the derivative of x², x³ or √x. You must show the limit notation and simplify the difference quotient fully.

在 Edexcel 考试中，第一原理题通常要求证明标准结果，例如 x²、x³ 或 √x 的导数。你必须展示极限符号并完整化简差商。

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