📚 A-Level CIE Mathematics: Calculus Basics – Exam Essentials | A-Level CIE 数学：微积分基础 考点精讲
Mastering the fundamentals of differentiation and integration is the cornerstone of success in CIE A-Level Mathematics (9709). This guide distills the key concepts, standard techniques, and common pitfalls into clear, paired explanations. Whether you are just starting Pure Mathematics 1 or revising for the final exam, these essentials will sharpen your skills and boost your confidence.
掌握微分与积分的基础知识是攻克 CIE A-Level 数学（9709）的基石。本文提炼核心概念、标准方法和易错陷阱，以清晰的中英对照阐释。无论你刚接触 Pure Mathematics 1 还是备考冲刺，这些要点都将打磨技巧、增强信心。
1. The Derivative from First Principles | 从第一性原理求导数
The derivative of a function f(x) is defined as f'(x) = limh→0 [f(x+h) – f(x)] / h. This limit gives the gradient of the tangent at any point. In CIE exams, you may be asked to prove the derivative of a simple function like x² or 1/x using first principles.
函数 f(x) 的导数定义为 f'(x) = limh→0 [f(x+h) – f(x)] / h。该极限给出任意点处切线的斜率。CIE 考试中可能会要求你用第一性原理证明简单函数（如 x² 或 1/x）的导数公式。
For f(x) = x², expand (x+h)² = x² + 2xh + h², then subtract f(x) to get 2xh + h². Dividing by h and taking the limit yields f'(x) = 2x.
以 f(x) = x² 为例，展开 (x+h)² = x² + 2xh + h²，减去 f(x) 得 2xh + h²，除以 h 后取极限即得 f'(x) = 2x。
2. Power Rule and Basic Differential Rules | 幂函数求导法则与基本微分规则
The power rule is the most fundamental tool: for any real constant n, d/dx (xⁿ) = n xⁿ⁻¹. It applies to positive integers, fractions (like √x = x^{1/2}), and negative powers (like 1/x² = x⁻²).
幂函数求导法则是基本工具：对于任意实常数 n，d/dx (xⁿ) = n xⁿ⁻¹。它适用于正整数、分数（如 √x = x^{1/2}）以及负指数（如 1/x² = x⁻²）。
The derivative of a constant is zero. For sums and differences, differentiate term by term: d/dx (f ± g) = f’ ± g’. Multiplying by a constant gives d/dx (k f) = k f’.
常数的导数为零。对于和或差，逐项求导：d/dx (f ± g) = f’ ± g’。乘以常数时，d/dx (k f) = k f’。
For example, differentiate y = 5x³ – 2x + 4: dy/dx = 15x² – 2.
例如，对 y = 5x³ – 2x + 4 求导：dy/dx = 15x² – 2。
3. The Chain Rule for Composite Functions | 复合函数的链式法则
The chain rule handles functions of the form y = (ax + b)ⁿ, y = e^{kx}, y = sin(kx), etc. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
链式法则处理形如 y = (ax + b)ⁿ、y = e^{kx}、y = sin(kx) 等函数。若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
For y = (3x – 5)⁴, let u = 3x – 5, so y = u⁴. Then dy/du = 4u³, du/dx = 3, giving dy/dx = 12(3x – 5)³.
对于 y = (3x – 5)⁴，令 u = 3x – 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 3，最终 dy/dx = 12(3x – 5)³。
In CIE, always remember to multiply by the derivative of the inner function. Common errors include forgetting to multiply by the coefficient of x.
在 CIE 考试中，永远记得乘以内层函数的导数。常见错误是忘记乘以 x 的系数。
4. The Product Rule and Quotient Rule | 乘积法则与商式法则
When two functions multiply, use the product rule: d/dx (u v) = u’ v + u v’. When one function divides another, use the quotient rule: d/dx (u/v) = (u’ v – u v’) / v².
两个函数相乘时使用乘积法则：d/dx (u v) = u’ v + u v’。一函数除以另一函数时使用商式法则：d/dx (u/v) = (u’ v – u v’) / v²。
For y = x² sin x, let u = x², v = sin x. Then u’ = 2x, v’ = cos x, so dy/dx = 2x sin x + x² cos x.
对于 y = x² sin x，令 u = x²，v = sin x。则 u’ = 2x，v’ = cos x，得 dy/dx = 2x sin x + x² cos x。
For y = (x + 1)/(x – 2), with u = x + 1, v = x – 2: dy/dx = [(1)(x – 2) – (x + 1)(1)] / (x – 2)² = –3/(x – 2)².
对于 y = (x + 1)/(x – 2)，令 u = x + 1，v = x – 2：dy/dx = [(1)(x – 2) – (x + 1)(1)] / (x – 2)² = –3/(x – 2)²。
5. Derivatives of Exponentials, Logarithms, and Trigonometric Functions | 指数、对数与三角函数的导数
Memorise these standard derivatives: d/dx (eˣ) = eˣ; d/dx (eᵏˣ) = k eᵏˣ. For natural log, d/dx (ln x) = 1/x for x > 0. For sine and cosine, d/dx (sin x) = cos x; d/dx (cos x) = –sin x.
熟记以下标准导数：d/dx (eˣ) = eˣ；d/dx (eᵏˣ) = k eᵏˣ。自然对数 d/dx (ln x) = 1/x（x > 0）。正弦与余弦：d/dx (sin x) = cos x；d/dx (cos x) = –sin x。
For tan x, remember d/dx (tan x) = sec² x. These often appear with the chain rule, e.g., d/dx (sin(2x)) = 2 cos(2x).
对于 tan x，记住 d/dx (tan x) = sec² x。它们常与链式法则结合，例如 d/dx (sin(2x)) = 2 cos(2x)。
In CIE Pure Mathematics 2 and 3, you also need d/dx (aˣ) = aˣ ln a and d/dx (logₐ x) = 1/(x ln a).
在 CIE Pure Mathematics 2 和 3 中，还需掌握 d/dx (aˣ) = aˣ ln a 以及 d/dx (logₐ x) = 1/(x ln a)。
6. Tangents, Normals, and Increasing/Decreasing Functions | 切线与法线、递增递减函数
The gradient of a curve at x = a is m = f'(a). The equation of the tangent is y – f(a) = f'(a)(x – a). The normal is perpendicular: its gradient is –1/f'(a) (provided f'(a) ≠ 0).
曲线在 x = a 处的斜率为 m = f'(a)。切线方程为 y – f(a) = f'(a)(x – a)。法线与之垂直：其斜率为 –1/f'(a)（若 f'(a) ≠ 0）。
A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0. You may be asked to find intervals of increase/decrease by solving f'(x) > 0 or f'(x) < 0.
一阶导数 f'(x) > 0 时函数递增，f'(x) < 0 时递减。考题可能要求通过解不等式 f'(x) > 0 或 f'(x) < 0 来找出递增或递减区间。
7. Stationary Points and the Second Derivative Test | 驻点与二阶导数判定法
Stationary points occur where dy/dx = 0. To classify them, compute the second derivative d²y/dx². If d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, a local maximum. If it equals zero, use the first derivative test (check sign changes).
驻点出现在 dy/dx = 0 处。分类方法：计算二阶导数 d²y/dx²。若 d²y/dx² > 0，则该点为极小值点；若 d²y/dx² < 0，则为极大值点。若等于零，则需用一阶导数符号变化判断。
For y = x³ – 3x, dy/dx = 3x² – 3. Setting to zero gives x = ±1. The second derivative d²y/dx² = 6x, so at x = 1 it is 6 > 0 (min), at x = –1 it is –6 < 0 (max).
以 y = x³ – 3x 为例，dy/dx = 3x² – 3，令其为零得 x = ±1。二阶导数 d²y/dx² = 6x，在 x = 1 处为 6 > 0（极小值），在 x = –1 处为 –6 < 0（极大值）。
8. Indefinite Integration: The Reverse of Differentiation | 不定积分：微分的逆运算
Integration reverses differentiation. The general rule is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ –1. The constant of integration C must be included. For n = –1, ∫ 1/x dx = ln|x| + C.
积分是微分的逆运算。一般规则为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ –1。积分常数 C 必须写上。当 n = –1 时，∫ 1/x dx = ln|x| + C。
Standard integrals also include ∫ eˣ dx = eˣ + C, ∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C.
标准积分还包括 ∫ eˣ dx = eˣ + C，∫ sin x dx = –cos x + C，∫ cos x dx = sin x + C。
For example, ∫ (4x³ – 2x + 1) dx = x⁴ – x² + x + C.
例如，∫ (4x³ – 2x + 1) dx = x⁴ – x² + x + C。
9. Definite Integration and Area Under a Curve | 定积分与曲线下方面积
A definite integral ∫ₐᵇ f(x) dx gives the exact area between the curve y = f(x) and the x-axis from x = a to x = b, provided the curve is entirely above the x‑axis on that interval. Evaluate using F(b) – F(a), where F(x) is an antiderivative of f(x).
定积分 ∫ₐᵇ f(x) dx 给出曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的精确面积，要求该区间上曲线完全位于 x 轴上方。计算方式为 F(b) – F(a)，其中 F(x) 是 f(x) 的一个原函数。
If the curve crosses the x‑axis, split the integral into sections where f(x) is positive and negative, taking absolute values of the negative parts to find the total area.
若曲线穿过 x 轴，需将积分分段处理，分为 f(x) 为正和负的区间，并对负值部分取绝对值以求得总面积。
Common mistake: forgetting to handle negative areas properly — always sketch the curve first in area problems.
常见错误：未正确处理负面积——做面积相关题目时应始终先画出草图。
10. Integration by Reverse Chain Rule and Substitution | 逆链式法则与代换积分法
The reverse chain rule handles integrals like ∫ f'(g(x)) g'(x) dx = f(g(x)) + C. In practice, recognise forms such as ∫ (ax + b)ⁿ dx or ∫ sin(ax + b) dx and adjust by dividing by the coefficient of x.
逆链式法则处理形如 ∫ f'(g(x)) g'(x) dx = f(g(x)) + C 的积分。实际做题时，识别 ∫ (ax + b)ⁿ dx 或 ∫ sin(ax + b) dx 等形式，并除以 x 的系数进行调整。
For example, ∫ (2x + 3)⁵ dx = (1/2) × (2x + 3)⁶ / 6 + C = (2x + 3)⁶ / 12 + C.
例如，∫ (2x + 3)⁵ dx = (1/2) × (2x + 3)⁶ / 6 + C = (2x + 3)⁶ / 12 + C。
In CIE P3, formal substitution (u = g(x)) is tested. Always express dx in terms of du and change limits in definite integrals.
在 CIE P3 中，会考查正式的代换法（u = g(x)）。务必用 du 表示 dx，并在定积分中更换积分上下限。
11. Connected Rates of Change | 关联变化率
When a variable y changes with respect to t and y is related to x, use the chain rule: dy/dt = (dy/dx) × (dx/dt). This is key in problems involving volume, radius, and time.
当变量 y 随 t 变化且 y 与 x 相关时，使用链式法则：dy/dt = (dy/dx) × (dx/dt)。这在涉及体积、半径与时间的问题中至关重要。
Example: a spherical balloon inflates so that its radius r increases at a constant rate dr/dt = 0.5 cm/s. The volume V = 4/3 π r³, so dV/dt = 4π r² × dr/dt. At r = 10 cm, dV/dt = 4π(100)(0.5) = 200π cm³/s.
例：球形气球充气时，半径 r 以恒定速率 dr/dt = 0.5 cm/s 增加。体积 V = 4/3 π r³，故 dV/dt = 4π r² × dr/dt。当 r = 10 cm 时，dV/dt = 4π(100)(0.5) = 200π cm³/s。
12. Exam Technique and Common Pitfalls | 考试技巧与常见误区
Always simplify your answer before applying the product or quotient rule if possible. For differentiation, write down all steps clearly to avoid sign errors. For integration, never forget the + C for indefinite integrals. In area problems, check where the curve meets the x‑axis before setting up definite integrals.
在应用乘积或商式法则前，尽可能先化简表达式。求导时，逐步写出清晰过程以避免符号错误。积分时，不定积分务必写上 + C。面积问题中，在设定积分前先检查曲线与 x 轴的交点。
When substituting limits, be meticulous with brackets, especially with negative numbers. Use a high‑lighter in the exam to mark key instructions like ‘exact value’, ‘simplify’, or ‘use substitution u = …‘.
代入上下限时，务必小心括号，尤其是涉及负数时。考试时可用高亮笔标记关键指令，如“精确值”、“化简”或“使用代换 u = …”。
Finally, practise past paper questions regularly. CIE often recycles problem types with slight variations. The more familiar you are with the pattern, the faster and more accurate your solutions will become.
最后，定期练习历年真题。CIE 常将题型稍作变化后重复考查。对题目模式越熟悉，解题就会越快、越准。
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