📚 Differentiation for Edexcel A-Level Maths | Edexcel A-Level 数学微分法
Differentiation is one of the core topics in the Edexcel A-Level Mathematics syllabus, underpinning work on gradients, rates of change, curve sketching and optimisation. This revision guide covers the key rules, techniques and types of questions you can expect in the exam.
微分是 Edexcel A-Level 数学大纲的核心主题之一，支撑着斜率、变化率、曲线作图和最优化等内容。本复习指南涵盖考试中可能出现的核心法则、技巧和题型。
1. What Is Differentiation? | 什么是微分？
Differentiation measures the rate at which one quantity changes with respect to another. For a curve y = f(x), the derivative dy/dx gives the gradient of the tangent at any point.
微分用来度量一个量相对于另一个量的变化率。对于曲线 y = f(x)，导数 dy/dx 给出任意一点切线的斜率。
The process is sometimes called ‘finding the gradient function’ because it produces a new function that can evaluate the slope at different x-values.
这一过程有时被称为 ‘求斜率函数’，因为它产生一个新函数，可以在不同的 x 值处计算斜率。
2. First Principles | 第一原理
The formal definition of the derivative from first principles is:
导数的第一原理定义是：
f'(x) = lim(h→0) [f(x+h) – f(x)] / h
Although you may not be asked to prove every derivative from first principles, Edexcel expects you to understand this limit definition and use it for simple polynomials.
虽然不要求用第一原理证明每个导数，但 Edexcel 希望考生理解这一极限定义，并能对简单多项式使用它。
For example, if f(x) = x², expanding f(x+h) = x² + 2xh + h² gives the limit as 2x, so f'(x) = 2x.
例如，如果 f(x) = x²，展开 f(x+h) = x² + 2xh + h² 后取极限得到 2x，因此 f'(x) = 2x。
3. Power Rule and Basic Derivatives | 幂法则与基本导数
For any real power n, the power rule states that d/dx (xⁿ) = n xⁿ⁻¹.
对于任意实数指数 n，幂法则指出 d/dx (xⁿ) = n xⁿ⁻¹。
You should also memorise the derivatives of eˣ, ln x, sin x, cos x and tan x.
你还应该熟记 eˣ、ln x、sin x、cos x 和 tan x 的导数。
The following table summarises the key derivatives.
下表总结了主要导数。
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
4. Product Rule and Quotient Rule | 乘法法则与除法法则
When differentiating a product of two functions, use the product rule:
当对两个函数的乘积求导时，使用乘法法则：
d/dx (uv) = u dv/dx + v du/dx
For quotients, use the quotient rule:
对于商，使用除法法则：
d/dx (u/v) = (v du/dx – u dv/dx) / v²
Pay attention to the order in the quotient rule: the denominator times the derivative of the numerator comes first.
注意除法法则中的顺序：分母乘以分子的导数在前。
5. Chain Rule | 链式法则
The chain rule is used for composite functions y = f(g(x)).
链式法则用于复合函数 y = f(g(x))。
dy/dx = dy/du × du/dx
For example, if y = (3x² + 1)⁵, let u = 3x² + 1, then dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴.
例如，如果 y = (3x² + 1)⁵，设 u = 3x² + 1，则 dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴。
You can also apply the chain rule mentally for linear inner functions such as sin(2x) → 2 cos(2x).
对于线性内层函数，如 sin(2x) → 2 cos(2x)，你也可以直接使用链式法则。
6. Differentiating Trigonometric, Exponential and Logarithmic Functions | 三角、指数和对数函数的微分
Common trigonometric derivatives include d/dx (sin kx) = k cos kx and d/dx (cos kx) = -k sin kx.
常见三角函数的导数包括 d/dx (sin kx) = k cos kx 和 d/dx (cos kx) = -k sin kx。
For exponential functions, d/dx (eᵏˣ) = k eᵏˣ, and for natural logs, d/dx (ln(kx)) = 1/x.
对于指数函数，d/dx (eᵏˣ) = k eᵏˣ；对于自然对数，d/dx (ln(kx)) = 1/x。
Be careful with trigonometric powers: d/dx (sin² x) = 2 sin x cos x by the chain rule.
小心三角函数的幂次：根据链式法则，d/dx (sin² x) = 2 sin x cos x。
7. Stationary Points and
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