Mastering Differentiation for Edexcel A-Level Pure Mathematics | 掌握 Edexcel A-Level 纯数学微分

📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | 掌握 Edexcel A-Level 纯数学微分

In Edexcel A-Level Mathematics, differentiation is one of the most heavily examined topics in Pure Mathematics. It underpins rates of change, gradient calculations, curve sketching, optimisation and many real-world modelling problems. This revision guide breaks down the key techniques, notation and common question types you need to master.

在 Edexcel A-Level 数学中，微分是纯数学部分考查频率最高的主题之一。它支撑着变化率、斜率计算、曲线作图、优化问题以及许多现实建模问题。本复习指南将拆解你需要掌握的关键技巧、符号和常见题型。

1. The Gradient Function and First Principles | 梯度函数与第一原理

The derivative of a function y = f(x) measures the gradient of the tangent to the curve at any point. It is defined as the limit of the average rate of change as the interval h tends to zero:

函数 y = f(x) 的导数衡量曲线上任意一点处切线的斜率。它被定义为当区间 h 趋于零时平均变化率的极限：

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

For example, when f(x) = x², expanding (x + h)² − x² gives 2xh + h², and after division by h the limit becomes 2x.

例如，当 f(x) = x² 时，展开 (x + h)² − x² 得到 2xh + h²，除以 h 后取极限得到 2x。

2. Standard Derivatives and Basic Rules | 标准导数与基本法则

For Edexcel A-Level, you must know the standard results for powers, trigonometric functions, exponentials and logarithms: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ, and d/dx (ln x) = 1/x. The constant multiple rule and sum rule allow you to differentiate term by term.

在 Edexcel A-Level 中，你必须掌握幂函数、三角函数、指数函数和对数函数的标准导数：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x。常数倍法则和加法法则使你能够逐项求导。

For example, d/dx (4x³ − 2 sin x + 5) = 12x² − 2 cos x.

例如，d/dx (4x³ − 2 sin x + 5) = 12x² − 2 cos x。

3. The Chain Rule | 链式法则

The chain rule is used when one function is inside another. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. In words, differentiate the outer function, keep the inner expression unchanged, and then multiply by the derivative of the inner expression.

链式法则用于一个函数嵌套在另一个函数内部的情况。如果 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。换句话说，先对外层函数求导，保持内层表达式不变，然后乘以内层表达式的导数。

Example: y = (3x² + 5)

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

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