📚 Mixed Exercise 12: Differentiation | 混合练习12：微分
Mixed Exercise 12 in the Edexcel AS/A Level Pure Mathematics course consolidates the first major calculus topic: differentiation. It tests first principles, polynomial derivatives, tangents and normals, stationary points, increasing and decreasing functions, second derivatives, and basic modelling. This bilingual revision guide walks through the key methods and common exam traps.
Edexcel AS/A Level 纯数学课程中的混合练习12 巩固了第一个重要微积分主题：微分。它考查第一性原理、多项式求导、切线与法线、驻点、增函数与减函数、二阶导数以及基本建模。本双语复习指南梳理关键方法和常见考试陷阱。
1. Definition and First Principles | 定义与第一性原理
The derivative of f(x) at a point measures the instantaneous rate of change of y with respect to x. It is defined as the limit of the average rate of change as h tends to 0.
函数 f(x) 在某一点的导数衡量 y 对 x 的瞬时变化率。它定义为当 h 趋向 0 时平均变化率的极限。
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
For example, if f(x) = x², then f(x+h) = (x+h)², so the limit becomes:
例如，若 f(x) = x²，则 f(x+h) = (x+h)²，因此极限变为：
f'(x) = lim(h→0) [(x+h)² − x²] / h = lim(h→0) (2xh + h²) / h = 2x
In Mixed Exercise 12, first-principles questions usually involve x², x³ or simple polynomials. Always expand brackets before simplifying.
在混合练习12中，第一性原理题通常涉及 x²、x³ 或简单多项式。化简前一定要先展开括号。
2. Differentiation Rules for Polynomials | 多项式求导法则
In Mixed Exercise 12, most functions are polynomials or simple algebraic terms. The key rules are the constant rule, the power rule, the constant multiple rule, and the sum rule.
在混合练习12中，大多数函数是多项式或简单代数项。关键法则是常数法则、幂法则、常数倍法则与和法则。
| Rule | 法则 | Example | 例子 |
|---|---|
| d/dx (c) = 0 | y = 5 ⇒ dy/dx = 0 |
| d/dx (xⁿ) = n xⁿ⁻¹ | y = x⁴ ⇒ dy/dx = 4x³ |
| d/dx (a xⁿ) = a n xⁿ⁻¹ | y = 3x⁵ ⇒ dy/dx = 15x⁴ |
| d/dx [f(x) + g(x)] = f'(x) + g'(x) | y = x³ + x² ⇒ dy/dx = 3x² + 2x |
Always simplify before differentiating, such as rewriting 1/x² as x⁻² and √x as x^½. This reduces sign and fraction errors.
求导前一定要先化简，例如将 1/x² 写成 x⁻²，将 √x 写成 x^½。这样可以减少符号和分数错误。
3. Tangents and Normals | 切线与法线
The gradient of the tangent to y = f(x) at x = a is m = f' uation of the tangent is y − y₁ = m(x − x₁). The normal has gradient −1/m, provided m ≠ 0.
曲线 y = f(x) 在 x = a 处切线的梯度为 m = f'(a)。切线方程为 y − y₁ = m(x − x₁)。法线梯度为 −1/m，前提是 m ≠ 0。
Worked example: Find the tangent and normal to y = x³ − 2x at x = 1.
例题：求 y = x³ − 2x 在 x = 1 处的切线与法线。
y(1) = 1³ − 2(1) = −1, y' = 3x² − 2, m = 3(1)² − 2 = 1
Tangent: y − (−1) = 1(x − 1) ⇒ y = x − 2. Normal: gradient = −1 ⇒ y + 1 = −1(x − 1) ⇒ y = −x.
切线：y − (−1) = 1(x − 1) ⇒ y = x − 2。法线：梯度 = −1 ⇒ y + 1 = −1(x − 1) ⇒ y = −x。
4. Stationary Points and Classification | 驻点及其分类
A stationary point occurs when f'(x) = 0. Solve f'(x) = 0 to find x-coordinates, then substitute into y = f(x) to find the y-coordinates.
驻点出现在 f'(x) = 0 时。解 f'(x) = 0 求出 x 坐标，再代入 y = f(x) 求出 y 坐标。
The second derivative classifies the stationary point:
二阶导数可对驻点进行分类：
| Condition | 条件 | Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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