Differentiation: Rules, Tangents, and Stationary Points | 微分：法则、切线与驻点

📚 Differentiation: Rules, Tangents, and Stationary Points | 微分：法则、切线与驻点

Differentiation is one of the core skills in the Edexcel A Level Mathematics specification. It measures the rate of change of a function and underpins problems on gradients, tangents, normals, stationary points, and optimisation. This revision article covers the main rules, standard derivatives, and the most common exam applications.

微分是 Edexcel A Level 数学大纲的核心技能之一。它度量函数的变化率，是梯度、切线、法线、驻点和优化问题的基础。本篇复习文章涵盖主要求导法则、标准导数以及最常见的考试应用。

1. What Is Differentiation? | 什么是微分？

Differentiation finds the gradient of a curve at a point. If y = f(x), the derivative dy/dx or f'(x) gives the instantaneous rate of change of y with respect to x. Geometrically, dy/dx is the slope of the tangent to the curve.

微分求曲线在某一点的梯度。若 y = f(x)，导数 dy/dx 或 f'(x) 表示 y 关于 x 的瞬时变化率。从几何上看，dy/dx 是曲线切线的斜率。

In exams, you usually apply standard rules rather than the limit definition. However, the limit idea explains why the derivative represents a tangent slope as h approaches zero.

考试中通常直接应用标准法则，而不是极限定义。但极限思想解释了为什么导数在 h 趋近于零时表示切线斜率。

2. The Power Rule and Basic Notation | 幂法则与基本记号

If y = xⁿ, then dy/dx = n xⁿ⁻¹, where n is a real constant. This is the most frequently used rule in A Level differentiation.

若 y = xⁿ，则 dy/dx = n xⁿ⁻¹，其中 n 为实常数。这是 A Level 微分中最常用的法则。

d/dx (xⁿ) = n xⁿ⁻¹

Constants differentiate to zero: if y = k, then dy/dx = 0. A constant multiple is preserved: d/dx [k f(x)] = k f'(x).

常数的导数为零：若 y = k，则 dy/dx = 0。常数倍可以保留：d/dx [k f(x)] = k f'(x)。

3. Differentiating Polynomials Term by Term | 多项式逐项求导

Differentiate each term separately and add the results. For example, if y = 4x³ – 2x² + 5x – 7, then dy/dx = 12x² – 4x + 5.

逐项求导后相加。例如，若 y = 4x³ – 2x² +

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