📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | 精通 Edexcel A-Level 纯数学微分
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It measures the rate at which one quantity changes with respect to another and gives the gradient of a curve at any point.
微分是 Edexcel A-Level 纯数学中考查最多的主题之一。它衡量一个量相对于另一个量的变化率，并给出曲线上任意一点的斜率。
This guide brings together the key ideas from Pure Year 1 and Year 2, including rules, notation, stationary points and exam applications.
本指南汇集 Pure 1 和 Pure 2 的关键知识，包括法则、记号、驻点以及考试应用。
1. What Is Differentiation? | 什么是微分？
Differentiation gives the instantaneous rate of change of a function f(x) with respect to x. The result is called the derivative, written as f'(x) or dy/dx.
微分给出函数 f(x) 对 x 的瞬时变化率。所得结果称为导数，记作 f'(x) 或 dy/dx。
derivative at a point is the gradient of the tangent to the curve y = f(x) at that point.
从几何角度看，某一点的导数就是曲线 y = f(x) 在该点处切线的斜率。
2. First Principles and the Limit Concept | 第一原理与极限概念
The derivative can be defined from first principles using a limit of a gradient chord as the interval tends to zero.
导数可以从第一原理出发，用割线斜率在区间趋于零时的极限来定义。
f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
In exams you rarely need to prove results from first principles, but you should recognise this expression and know how to apply it to simple polynomials.
考试中很少要求用第一原理证明公式，但你应识别该表达式，并知道如何将其应用于简单多项式。
3. Power Rule and Basic Results | 幂法则与基本结果
For any real constant n, the derivative of xⁿ is n xⁿ⁻¹.
对于任意实常数 n，xⁿ 的导数为 n xⁿ⁻¹。
d/dx (xⁿ) = n xⁿ⁻¹
This rule works for rational powers, so expressions such as √x = x½ differentiate to ½ x⁻½.
该法则适用于有理指数，因此 √x = x½ 的导数为 ½ x⁻½。
an also differentiate constant multiples and sums term by term: d/dx [af(x) + bg(x)] = a f'(x) + b g'(x).
你还可以逐项微分常数倍和和差：d/dx [af(x) + bg(x)] = a f'(x) + b g'(x)。
| f(x) | f'(x) |
|---|---|
| x³ | 3x² |
| 5x⁴ | 20x³ |
| 2√x = 2x½ | x⁻½ |
| 3/x² = 3x⁻² | -6x⁻³ |
Always simplify a function into index form before differentiating.
微分前务必先将函数化简为指数形式。
4. Differentiating Standard Functions | 标准函数的微分
You must memorise the derivatives of common functions, especially for Edexcel Year 2 questions involving exponentials, logarithms and trigonometric functions.
你必须熟记常见函数的导数，尤其是 Edexcel 第二年试题中涉及的指数、对数和三角函数。
| f(x) | f'(x) |
|---|---|
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
These results are often combined with the chain rule, such as d/dx [e²ˣ] = 2e²ˣ and d/dx [
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