Differentiation Rules and Applications for Edexcel A-Level Maths | Edexcel A-Level数学：微分法则与应用

📚 Differentiation Rules and Applications for Edexcel A-Level Maths | Edexcel A-Level数学：微分法则与应用

In Edexcel A-Level Mathematics, differentiation is a core pure topic that underpins applications in areas such as stationary points, tangents, normals and modelling with rates of change. This revision guide covers the key differentiation rules and the most common ways they are tested in the exam.

在 Edexcel A-Level 数学中，微分是纯数部分的核心主题，为驻点、切线、法线以及变化率建模等内容打下基础。本复习指南涵盖关键微分法则以及考试中最常见的考查方式。

1. What Differentiation Measures | 微分衡量什么

The derivative dy/dx gives the instantaneous rate of change of y with respect to x. Geometrically, it represents the gradient of the tangent to the curve y = f(x) at any point.

导数 dy/dx 表示 y 关于 x 的瞬时变化率。从几何上看，它表示曲线 y = f(x) 在任意一点处切线的斜率。

Differentiation is therefore used whenever a question asks for a gradient, an equation of a tangent, a rate of increase, or a maximum/minimum value.

因此，当题目要求求斜率、切线方程、增长速度或最大值/最小值时，通常都要使用微分。

In Edexcel exam papers, the derivative is often connected to real contexts such as volume, temperature, distance or profit. Being able to interpret dy/dx in words is as important as calculating it correctly.

在 Edexcel 试卷中，导数经常与体积、温度、距离或利润等实际情境联系起来。能够用文字解释 dy/dx 的含义与正确计算它同样重要。

2. The Power Rule and Basic Results | 幂法则与基本结果

For any real constant n, the power rule states:

对于任意实数常数 n，幂法则为：

y = xⁿ → dy/dx = n xⁿ⁻¹

This is the first tool to apply when differentiating polynomial terms such as 3x⁴ − 5x² + 2x − 7. Differentiate term by term to obtain dy/dx = 12x³ − 10x + 2.

这是对 3x⁴ − 5x² + 2x − 7 这样的多项式各项求导时首先使用的工具。逐项求导可得 dy/dx = 12x³ − 10x + 2。

You should also memorise the standard derivatives for eᵏˣ, ln x, sin x, cos x and tan x because Edexcel papers often combine them with the chain, product or quotient rules.

你还应熟记 eᵏˣ、ln x、sin x、cos x 和 tan x 的标准导数，因为 Edexcel 试卷经常将它们与链式法则、乘法法则或除法法则结合考查。

y = f(x)	dy/dx	中文释义
eᵏˣ	k eᵏˣ	指数函数
ln x	1/x	自然对数
sin x	cos x	正弦函数
cos x	−sin x	余弦函数
tan x	sec² x	正切函数

These results form the foundation of nearly every differentiation question. When combined with the chain rule, they extend to more complicated expressions such as e⁻²ˣ or sin(5x + 1).

这些结果是几乎所有微分题的基础。当与链式法则结合时，它们可以推广到更复杂的表达式，例如 e⁻²ˣ 或 sin(5x + 1)。

3. The Chain Rule | 链式法则

The chain rule is used when one function is applied inside another. If y = f(u) and u = g(x), then:

链式法则用于一个函数嵌套在另一个函数中的情况。若 y = f(u) 且 u = g(x)，则：

dy/dx = dy/du × du/dx

For example, y = (3x² + 5)⁴ can be seen as y = u⁴ with u = 3x² + 5, giving dy/du = 4u³ and du/dx = 6x.

例如，y = (3x² + 5)⁴ 可看作 y = u⁴，其中 u = 3x² + 5，于是 dy/du = 4u³，du/dx = 6x。

Substituting back gives dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³. Leaving the inner function unchanged is an important part of writing the final answer clearly.

代回可得 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。让内层函数保持不变是清晰写出最终答案的重要部分。

Many Edexcel questions contain a ‘dy/dx = dy/du × du/dx’ structure hidden inside a product or quotient, so identifying the inner function is a crucial exam skill.

许多 Edexcel 题目将 dy/dx = dy/du × du/dx 的结构隐藏在乘法或除法函数中，因此识别内层函数是一项关键的考试技能。

4. The Product Rule | 乘法法则

When y = u v and both u and v are functions of x, the product rule gives:

当 y = u v 且 u、v 都是 x 的函数时，乘法法则为：

dy/dx = u dv/dx + v du/dx

The key is to keep the two factors intact in the first stage, then differentiate each one separately. For y = x² sin x, let u = x² and v = sin x, so du/dx = 2x and dv/dx = cos x.

关键是在第一步中保持两个因式不变，然后分别求导。对于 y = x² sin x，令 u = x²、v = sin x，则 du/dx = 2x，dv/dx = cos x。

Substituting gives dy/dx = x² cos x + 2x sin x. This is already simplified, but in other examples factorising may reveal the structure needed for the next part of the question.

代入可得 dy/dx = x² cos x + 2x sin x。这个答案已经化简，但在其他例子中，因式分解可能会揭示题目下一部分所需的结构。

It is often useful to factorise the final answer. Edexcel marking schemes accept both factorised and expanded forms, but factorised forms are easier to use when finding stationary points.

最后答案通常进行因式分解会更有用。Edexcel 评分方案既接受因式分解形式也接受展开形式，但在求驻点时因式分解形式更便于使用。

5. The Quotient Rule | 除法法则

If y = u/v where u and v are functions of x, then:

若 y = u/v 且 u、v 都是 x 的函数，则：

dy/dx = (v du/dx − u dv/dx) / v²

The order in the numerator matters: it is v du/dx minus u dv/dx, not the reverse. Start by writing u, v, du/dx and dv/dx separately before substituting into the formula.

分子的顺序很重要：是 v du/dx 减去 u dv/dx，不能颠倒。先把 u、v、du/dx、dv/dx 分别写出，再代入公式。

For y = x³ / ln x, set u = x³ and v = ln x. Then du/dx = 3x² and dv/dx = 1/x, so:

对于 y = x³ / ln x，令 u = x³、v = ln x，则 du/dx = 3x²，dv/dx = 1/x，因此：

dy/dx = [ln x × 3x² − x³ × 1/x] / (ln x)²

This simplifies to (3x² ln x − x²) / (ln x)². Always check whether the numerator can be factorised before carrying out further work such as solving dy/dx = 0.

可化简为 (3x² ln x − x²) / (ln x)²。在进行进一步计算（例如解 dy/dx = 0）之前，始终检查分子是否可以因式分解。

6. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative d²y/dx² is obtained by differentiating dy/dx again. It measures the rate of change of the gradient and tells you whether a curve is concave up or concave down.

二阶导数 d²y/dx² 是对 dy/dx 再次求

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Differentiation Rules and Applications for Edexcel A-Level Maths | Edexcel A-Level数学：微分法则与应用