📚 Mastering Differentiation for Edexcel A-Level Pure Maths | 掌握爱德思A-Level纯数学：微分技巧
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It underpins applications such as optimisation, curve sketching, kinematics and connected rates of change. Mastering the rules and their applications gives you a reliable source of marks across Paper 1 and Paper 2.
微分是爱德思 A-Level 纯数学中考查频率最高的主题之一。它是优化、曲线作图、运动学以及相关变化率等应用的基础。掌握微分法则及其应用，能够为你在 Paper 1 和 Paper 2 中稳定得分。
1. What Differentiation Means | 微分的意义
y/dx gives the gradient of the tangent at any point. This is why differentiation is used to find stationary points, velocities and marginal rates in real-world models.
微分衡量函数在某一点的瞬时变化率。对于曲线 y = f(x)，导数 dy/dx 表示任意一点处切线的斜率。因此，微分被用来求驻点、速度以及现实模型中的边际变化率。
In Edexcel exam questions, you should be able to interpret dy/dx in context, such as saying 'the rate of change of y with respect to x' and explaining what a negative gradient means.
在爱德思考题中，你应当能够在具体情境中解释 dy/dx 的含义，例如描述为“y 关于 x 的变化率”，并解释负斜率的实际意义。
2. Differentiation from First Principles | 从第一性原理出发
x)] / h. Edexcel often asks candidates to differentiate simple functions such as x², x³ or √x using this definition, before applying the power rule.
导数的正式定义是 f ′(x) = lim (h → 0) [f(x + h) − f(x)] / h。爱德思经常要求考生使用这一定义对 x²、x³ 或 √x 等简单函数求导，再应用幂法则。
A full first-principles solution must show the limit clearly, expand brackets correctly and cancel h before letting h tend to 0. Missing the limit notation is a common cause of lost marks.
完整的第一性原理解题必须清楚地写出极限，正确展开括号，并在令 h 趋于 0 之前约去 h。漏写极限符号是常见的失分原因。
3. Basic Differentiation Rules | 基本微分法则
rule extends to any real constant n, so it also covers terms like 1/x² written as x⁻² and √x written as x raised to the power ½.
幂法则指出，对于 f(x) = xⁿ，f ′(x) = n xⁿ⁻¹。该法则适用于任意实常数 n，因此也涵盖写作 x⁻² 的 1/x² 和写作 x 的 ½ 次方的 √x。
out and sums are differentiated term by term. For example, d/dx (3x⁴ − 5x + 2) = 12x³ − 5.
微分是线性的，因此常数因子可以提出，和式可以逐项求导。例如 d/dx (3x⁴ − 5x + 2) = 12x³ − 5。
| Function | Derivative |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| a xⁿ | a n xⁿ⁻¹ |
| f(x) ± g(x) | f ′(x) ± g ′(x) |
4. Differentiating Standard Functions | 标准函数的导数
Edexcel expects you to recall the derivatives of common functions: d/dx (eˣ) = eˣ d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, and d/dx (tan x) = sec² x.
爱德思要求你熟记常见函数的导数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，以及 d/dx (tan x) = sec² x。
For these standard results to be applied directly, the coefficient of x inside the function must be exactly 1. If the argument is more complicated, such as e²ˣ or sin 3x, the chain rule is needed.
要直接应用这些标准结果，函数中 x 的系数必须恰好为 1。如果自变量更复杂，例如 e²ˣ 或 sin 3x，就需要使用链式法则。Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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