A-Level Edexcel Maths: Differentiation Essentials | A-Level Edexcel 数学：微分核心要点

📚 A-Level Edexcel Maths: Differentiation Essentials | A-Level Edexcel 数学：微分核心要点

This guide focuses on the differentiation skills most frequently tested in Edexcel A-Level Mathematics, from basic rules to connected rates and optimisation. It is designed as a quick but thorough revision resource.

本指南聚焦 Edexcel A-Level 数学中最常考查的微分技能，从基本法则到相关变化率和优化问题。它可作为快速但全面的复习资源。

Work through each section, rewrite the featured formulas, and then attempt past-paper questions on the same topic before checking your solutions.

逐节学习，重写给出的公式，然后在核对答案前尝试同主题的历年真题。

1. Power Rule and Basic Derivatives | 幂函数法则与基本导数

The power rule is the foundation of most Edexcel differentiation questions. If y = xⁿ, then dy/dx = n xⁿ⁻¹ for any real constant n.

幂函数法则是大多数 Edexcel 微分题的基础。如果 y = xⁿ，则 dy/dx = n xⁿ⁻¹，其中 n 为任意实常数。

d/dx (xⁿ) = n xⁿ⁻¹

This rule also covers negative and fractional powers. For instance, y = 1/x² = x⁻² gives dy/dx = -2 x⁻³, and y = √x = x½ gives dy/dx = ½ x⁻½.

该法则也适用于负指数和分数指数。例如，y = 1/x² = x⁻² 可得 dy/dx = -2 x⁻³；y = √x = x½ 可得 dy/dx = ½ x⁻½。

You must also know the standard results 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.

你还必须掌握标准结论：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。

2. Tangent and Normal Lines | 切线与法线

At a point x = a, the gradient of a curve is the value of dy/dx at that point. This gives the gradient of the tangent line to the curve.

在点 x = a 处，曲线的斜率是该点 dy/dx 的值。它也是曲线在该点切线的斜率。

y – y₁ = m(x – x₁)

If the tangent gradient is m, the normal gradient is -1/m. The normal is perpendicular to the tangent and often appears in coordinate geometry questions.

如果切线斜率为 m，则法线斜率为 -1/m。法线与切线垂直，经常出现在坐标几何题中。

Always substitute the original point into the derivative to find m, then use the line equation to write the tangent or normal in the required form.

始终将原坐标代入导数求出 m，再使用直线方程将切线或法线写成题目要求的形式。

3. Increasing and Decreasing Functions | 递增与递减函数

A function is increasing on an interval when f'(x) > 0 and decreasing when f'(x) < 0. This sign test is often linked to stationary points and curve sketching.

当 f'(x) > 0 时，函数在区间上递增；当 f'(x) < 0 时递减。这种符号判断经常与驻点和曲线草图联系在一起。

Draw a sign diagram for f'(x) to show intervals of increase and decrease clearly. Edexcel examiners want the sign diagram to support your conclusion.

绘制 f'(x) 的符号图，可以清晰地表示递增和递减区间。Edexcel 阅卷人希望看到符号图来支持你的结论。

4. Stationary Points and Classification | 驻点及其分类

Stationary points occur

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