📚 Edexcel Maths: Mastering Differentiation | Edexcel 数学：微分 考点精讲
Differentiation is a fundamental pillar of Edexcel A-Level Mathematics, and it is one of the few topics that appears in every Pure paper. From basic power functions to implicit and parametric equations, mastery of this topic is essential for top exam performance. This guide covers every key technique, with step-by-step explanations, worked examples and the common pitfalls that cost students easy marks.
微分是 Edexcel A-Level 数学的基石，也是少数几乎出现在每一份 Pure 试卷中的主题。从基本的幂函数到隐函数和参数方程，掌握这一主题对于取得优异成绩至关重要。本指南涵盖每一个关键技巧，提供逐步讲解、典型例题，以及那些最容易让学生丢分的常见错误。
Everything below follows the assessment style of the Edexcel specification: a short piece of technique, then an exam-style application. Read the method, work the example yourself, then check the answer before moving on.
下文内容完全按照 Edexcel 考纲的考查方式来组织：先讲技巧，再给考试风格的运用。请先读方法，自己动手算一遍例题，再核对答案，然后进入下一节。
1. The Exam Context and What You Must Know | 考试背景与必备知识
In Edexcel A-Level Mathematics (9MA0), differentiation is examined in both Pure 1 and Pure 2. Pure 1 introduces the idea of the gradient function and the power rule; Pure 2 extends this to the chain rule, product rule, quotient rule, second derivatives, rates of change, implicit differentiation and parametric differentiation. Further Mathematics pushes further into inverse trigonometric functions, Maclaurin series and differential equations.
在 Edexcel A-Level 数学（9MA0）中，微分在 Pure 1 和 Pure 2 中都会考查。Pure 1 引入导函数的概念与幂法则；Pure 2 则扩展到链式法则、乘积法则、商法则、二阶导数、变化率、隐函数微分以及参数方程微分。进阶数学（Further Maths）会进一步涉及反三角函数、麦克劳林级数和微分方程。
The definition you must understand, not merely quote, is the limit definition of the derivative. It states that the derivative is the limiting value of the gradient of a chord as the chord shrinks to a tangent.
你必须真正理解、而不仅仅是背诵的定义是导数的极限定义：当割线不断收缩为切线时，割线斜率的极限值就是导数。
dy/dx = lim (Δy/Δx) as Δx → 0
Differentiation from first principles is usually asked for small positive integer powers of x, and occasionally for sin x and cos x. In most other questions you are expected to use the standard rules fluently and quickly.
一阶原理求导通常要求针对 x 的小正整数次幂，偶尔也会考查 sin x 与 cos x。在其余大多数题目中，你需要熟练、快速地运用标准法则。
2. Power Rule and Basic Derivatives | 幂法则与基本导数
⁻¹. This rule works for any real exponent n, including fractions and negatives. It is the single most used rule in the whole paper, so it must be automatic.
若 y = xⁿ，则导数为 dy/dx = n xⁿ⁻¹。该法则适用于任何实数指数 n，包括分数和负数。它是整份试卷中使用频率最高的法则，必须做到条件反射般熟练。
d/dx (xⁿ) = n xⁿ⁻¹
Constant multiples and sums: d/dx [c f(x)] = c f'(x) and d/dx [f(x) ± g(x)] = f'(x) ± g'x). In words, constants slide out of the derivative and terms differentiate independently.
常数倍与和差法则：d/dx [c f(x)] = c f'(x)，以及 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。也就是说，常数可以提到导数外面，各项可以分别求导。
. The power rule handles 4x³, the linear term −2x becomes −2, and the constant disappears.
示例：y = 4x³ − 2x + 7 → dy/dx = 12x² − 2。幂法则处理 4x³，一次项 −2x 变成 −2，常数项则消失。
Do not forget that the derivative of a constant is zero. A surprisingly large number of marks are lost each session simply because a student writes a constant term into the answer.
不要忘记常数的导数为零。每一次考试都有相当多的分数，仅仅因为学生在答案里保留了常数项而白白丢掉。
Before differentiating anything, rewrite it in index form. Roots and reciprocals become powers: √x = x^½, 1/x = x⁻¹, 1/x² = x⁻², and 5/√x = 5x−½. Only then apply the rule.
在求导之前，先把表达式改写为指数形式。根式和倒数都要化为幂：√x = x^½，1/x = x⁻¹，1/x² = x⁻²，5/√x = 5x−½。完成改写后再套用法则。
- y = √x = x^½ → dy/dx = ½ x−½ = 1/(2√x)
y = √x = x^½ → dy/dx = ½ x−½ = 1/(2√x) - y = 3/x² = 3x⁻² → dy/dx = −6x⁻³ = −6/x³
y = 3/x² = 3x⁻² → dy/dx = −6x⁻³ = −6/x³ - y = x → dy/dx = 1, and y = 5 → dy/dx = 0
y = x → dy/dx = 1，y = 5 → dy/dx = 0
3. Chain Rule | 链式法则
When a function is a composition y = f(g(x)), the chain rule states: dy/dx = f'(g(x)) · g' mes derivative of the inner function. This is the most frequently examined rule in Pure 2.
当函数为复合型 y = f(g(x))，链式法则为：dy/dx = f'(g(x)) · g'(x)。即外函数的导数乘以内函数的导数。这是 Pure 2 中考查频率最高的法则。
dy/dx = dy/du · du/dx
u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³.
例如 y = (3x² + 5)⁴，令 u = 3x² + 5。则 dy/du = 4u³，du/dx = 6x，所以 dy/dx = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³。
Common mistake: forgetting to multiply by the derivative of the inner function. If your answer for a composite function contains no trace of the inner derivative, it is almost certainly wrong.
常见错误：忘记乘以内函数的导数。如果复合函数的答案里完全看不到内层导数的影子，那几乎可以肯定答案是错的。
The chain rule also governs the standard composites you will meet again and again. Learn these three patterns and most chain-rule questions collapse into one line of working.
链式法则同样支配着你会反复遇到的几类标准复合形式。掌握下面三种模式，大多数链式法则题目都只剩一行运算。
- y = (ax + b)ⁿ → dy/dx = an(ax + b)ⁿ⁻¹
y = (ax + b)ⁿ → dy/dx = an(ax + b)ⁿ⁻¹ - y = eax + b → dy/dx = a eax + b
y = eax + b → dy/dx = a eax + b - y = ln(ax + b) → dy/dx = a/(ax + b)
y = ln(ax + b) → dy/dx = a/(ax + b)
e outside in: differentiate y = sin(ln(3x)) as cos(ln(3x)) · (1/(3x)) · 3 = cos(ln(3x))/x.
遇到更深的嵌套时，由外向内反复使用链式法则：对 y = sin(ln(3x)) 求导得 cos(ln(3x)) · (1/(3x)) · 3 = cos(ln(3x))/x。
4. Product Rule | 乘积法则
du/dx. A handy mnemonic: first times derivative of second, plus second times derivative of first. Write the two products down before simplifying, so you never lose a term.
若 y = u(x) v(x)，则 dy/dx = u dv/dx + v du/dx。容易记忆：第一个乘第二个的导数，加上第二个乘第一个的导数。先完整写出两项乘积，再化简，这样就不会漏项。
y' = u v' + v u'
Example: y = x² sin x. Let u = x² and v = sin x. Then u' = 2x and v' = cos x. So dy/dx = x² cos x + sin x · 2x, which may be written as x(x cos x + 2 sin x).
示例：y = x² sin x。令 u = x²，v = sin x。则 u' = 2x，v' = cos x。所以 dy/dx = x² cos x + sin x · 2x，也可整理为 x(x cos x + 2 sin x)。
Fully factorised answers are preferred in mark schemes and are essential in later parts of a question, for example when you must set the derivative equal to zero and solve.
评分标准更偏好完全因式分解的答案，而且在题目的后续小问中这往往是必需的，例如当你需要令导数为零并求解时。
to avoid the product rule. For y = (2x + 3)(x² − 1) either method works, but for y = x⁵ ex or y = x ln x, expanding is impossible and the product rule is the only route.
不要为了回避乘积法则而把括号乘开。对 y = (2x + 3)(x² − 1) 两种方法都可行，但对 y = x⁵ ex 或 y = x ln x，展开根本做不到，乘积法则是唯一出路。
Common mistake: differentiating only one factor. A second classic error is placing a minus sign where the product rule requires a plus sign. The product rule always has two positive signs between its terms.
常见错误：只对一个因子求导。第二个典型错误是把乘积法则中的加号写成减号。乘积法则的两项之间永远是加号。
5. Quotient Rule | 商法则
dx − u dv/dx)/v². The mnemonic most students use is: low d high minus high d low, over low squared. The order matters, because the numerator contains a subtraction.
若 y = u/v（v ≠ 0），则 dy/dx = (v du/dx − u dv/dx)/v²。常用的口诀是：分母乘分子的导数，减去分子乘分母的导数，整体除以分母的平方。顺序很重要，因为分子中有减法。
dy/dx = (v u' − u v') / v²
Example: y = (2x + 1)/(x − 3). Here u = 2x + 1, v = x − 3, u' = 2, v' Then dy/dx = [2(x − 3) − (2x + 1)(1)]/(x − 3)² = (2x − 6 − 2x − 1)/(x − 3)² = −7/(x − 3)².
示例：y = (2x + 1)/(x − 3)。此处 u = 2x + 1，v = x − 3，u' = 2，v' = 1。则 dy/dx = [2(x − 3) − (2x + 1)(1)]/(x − 3)² = (2x − 6 − 2x − 1)/(x − 3)² = −7/(x − 3)²。
Notice how much information the answer carries: it is always negative, confirming the function is decreasing everywhere on its domain, and it tends to zero as x grows, showing the curve flattens towards a horizontal asymptote.
注意这个答案携带了大量信息：它恒为负，说明该函数在其定义域上单调递减；当 x 增大时它趋于零，说明曲线逐渐趋近于一条水平渐近线。
, writing y = x³(x + 1)⁻¹ avoids the quotient rule altogether and often simplifies faster.
一个实用的替代方法，是把商改写为带负指数的乘积，然后使用乘积法则。例如 y = x³/(x + 1) 可写成 y = x³(x + 1)⁻¹，从而完全避开商法则，通常化简更快。
Common mistake: reversing the numerator. If you compute u v' − v u' instead of v u' − u v', every sign in the answer flips. Common mistake: forgetting to square the denominator, or differentiating the denominator and leaving it unsquared.
常见错误：把分子顺序颠倒。若你算的是 u v' − v u' 而不是 v u' − u v'，答案中所有符号都会相反。另一个常见错误是忘记对分母平方，或对分母求导后却没有平方。
6. Standard Derivatives You Cannot Afford to Forget | 必须牢记的标准导数
The formulae booklet supplies a table of standard derivatives, but it does not supply the chain, product or quotient rules themselves. You must therefore memorise both the table and the rules for combining it.
公式手册中提供了标准导数表，但并不会给出链式、乘积与商法则本身。因此你既要背下表，也要熟记把它们组合起来使用的规则。
| f(x) | f'(x) | Composite form | 复合形式 |
|---|---|---|---|
| sin x | cos x | a cos(ax + b) | a cos(ax + b) |
| cos x | −sin x | −a sin(ax + b) | −a sin(ax + b) |
| tan x | sec² x | a sec²(ax + b) | a sec²(ax + b) |
| ex | ex | k ekx | k ekx |
| ln x | 1/x | f'(x)/f(x) | f'(x)/f(x) |
Two results deserve special attention. First, differentiating ln(f(x)) gives f'(x)/f(x), a pattern that appears in every paper. Second, differentiating ax gives ax ln a, which follows from writing ax as ex ln a.
有两个结果需要特别关注。第一，对 ln(f(x)) 求导得到 f'(x)/f(x)，这个模式几乎每份试卷都会出现。第二，对 ax 求导得到 ax ln a，它来自把 ax 写成 ex ln a 后的链式法则。
Watch the angles. In A-Level Mathematics all trigonometric derivatives assume x is measured in radians. Differentiating sin x in degrees requires an extra factor of π/180, which is why degree-mode calculus is never examined.
注意角的单位。在 A-Level 数学中，所有三角函数的导数都默认 x 以弧度为单位。若以度为单位，对 sin x 求导需要额外乘以 π/180，这正是考试从不以角度制考查微积分的原因。
7. Second Derivatives and Notation | 二阶导数与记号
The derivative of the derivative is the second derivative, written d²y/dx² or f''(x). It measures how the gradient itself is changing, and it is the standard tool for classifying stationary points.
对导数再求一次导，就得到二阶导数，记作 d²y/dx² 或 f''(x)。它衡量的是斜率本身如何变化，是判断驻点性质的标准
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