IB AQA Maths: Calculus Fundamentals Key Points | IB AQA 数学：微积分基础 考点精讲

📚 IB AQA Maths: Calculus Fundamentals Key Points | IB AQA 数学：微积分基础 考点精讲

Calculus is a cornerstone of higher mathematics, providing tools to analyse change and accumulation. For both IB (Analysis & Approaches) and AQA A-level Maths, a solid grasp of differentiation and integration fundamentals is essential for success. This article revisits the key concepts, rules and common exam pitfalls, pairing each English explanation with its Chinese equivalent to support bilingual learners.

微积分是高等数学的基石，提供了分析变化与累积的工具。无论是 IB（分析与方法）还是 AQA A-level 数学，扎实掌握微分与积分的基础知识都是取得好成绩的关键。本文梳理核心概念、法则与常见考试易错点，并为每条英文解说配搭中文译文，以支援双语学习者。

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1. Definition of the Derivative | 导数的定义

The derivative of a function f at a point x is the limit of the difference quotient as h approaches zero. It measures the instantaneous rate of change of f with respect to x.

函数 f 在点 x 处的导数是当 h 趋近于 0 时差商的极限。它度量了 f 关于 x 的瞬时变化率。

f '(x) = lim_h→₀ [f(x + h) – f(x)] / h

the tangent line to the curve y = f(x) at the point (x, f(x)), provided the limit exists.

几何上，若该极限存在，它给出了曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。

first principles using this limit definition. Practice with simple functions like f(x) = x².

在 IB 和 AQA 考试中，你可能会被要求利用此极限定义从第一原理求导。建议先用 f(x) = x² 等简单函数进行练习。

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2. Basic Differentiation Rules | 基本求导法则

Memorising the standard derivatives saves time. The power rule states: if f(x) = xⁿ, then f '(x) = n x^n–1.

熟记标准导数能节省时间。幂函数法则为：若 f(x) = xⁿ，则 f '(x) = n x^n–1。

d/dx (xⁿ) = n x^n–1

onstants differentiate to zero: d/dx (c) = 0. The linearity of differentiation allows term‑by‑term work: d/dx [a f(x) + b g(x)] = a f '(x) + b g '(x).

常数的导数为零：d/dx (c) = 0。微分的线性性质允许逐项求导：d/dx [a f(x) + b g(x)] = a f '(x) + b g '(x)。

Key function derivatives to know:

需要掌握的关键函数导数：

• d/dx (e^x) = e^x   导数 d/dx (e^x) = e^x
• d/dx (ln x) = 1/x   导数 d/dx (ln x) = 1/x
• d/dx (sin x) = cos x   导数 d/dx (sin x) = cos x
• d/dx (cos x) = –sin x   导数 d/dx (cos x) = –sin x
• d/dx (tan x) = sec² x   导数 d/dx (tan x) = sec² x

Always check whether the variable is in radians; trigonometric derivatives assume radian measure.

务必确认角度单位是弧度；三角函数的导数公式均以弧度制为前提。

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3. The Chain Rule | 链式法则

rule differentiates composite functions. If y = f(g(x)), then the derivative is dy/dx = f '(g(x)) · g '(x).

链式法则用于复合函数求导。若 y = f(g(x))，则导数为 dy/dx = f '(g(x)) · g '(x)。

In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx).

用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) · (du/dx)。

Example: Differentiate y = (3x² + 1)⁵. Here u = 3x² + 1, so dy/du = 5u⁴, du/dx = 6x, giving dy/dx = 5(3x² + 1)⁴ · 6x.

示例：求 y = (3x² + 1)⁵ 的导数。设 u = 3x² + 1，则 dy/du = 5u⁴，du/dx = 6x，得 dy/dx = 5(3x² + 1)⁴ · 6x。

Remember to multiply by the derivative of the inner function – this is the most common mistake in exams.

切记要乘以内层函数的导数——这是考试中最常见的错误。

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4. Product and Quotient Rules | 乘积法则与商法则

For the product of two functions, the product rule states: if y = u v, then dy/dx = u' v + u v'.

对于两个函数的乘积，乘积法则为：若 y = u v，则 dy/dx = u' v + u v'。

d/dx (u v) = u' v + u v'

For the quotient, the quotient rule gives: if y = u / v, then dy/dx = (u' v – u v') / v².

对于分式，商法则为：若 y = u / v，则 dy/dx = (u' v – u v') / v²。

A common mnemonic is "low d-high minus high d-low, over low squared". Be careful with the minus sign – it appears before u v', not after.

常见的记忆口诀是“分母乘分子导数减分子乘分母导数，除以分母的平方”。注意减号在 u v' 之前而非之后。

Example: For y = x² sin x, u = x², v = sin x ⇒ dy/dx = 2x sin x + x² cos x.

示例：y = x² sin x，令 u = x², v = sin x，则 dy/dx = 2x sin x + x² cos x。

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5. Implicit Differentiation | 隐函数微分

When a relation between x and y is given implicitly (e.g. x² + y² = 1), differentiate both sides with respect to x, treating y as a function of x.

当 x 与 y 的关系以隐式给出（例如 x² + y² = 1），两边同时对 x 求导，将 y 视为 x 的函数。

For every y-term, apply the chain rule: d/dx (yⁿ) = n y^n–1 (dy/dx). After differentiating, rearrange to solve for dy/dx.

对每个含 y 的项施加链式法则：d/dx (yⁿ) = n y^n–1 (dy/dx)。求导后，移项解出 dy/dx。

Example: x² + y² = 1 → 2x + 2y (dy/dx) = 0 → dy/dx = –x/y.

示例：x² + y² = 1 → 2x + 2y (dy/dx) = 0 → dy/dx = –x/y。

is essential for finding gradients of curves that cannot be easily expressed as y = f(x).

对于难以表示为显函数 y = f(x) 的曲线，隐函数微分是求切线梯度的基本方法。

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6. Higher-Order Derivatives | 高阶导数

The second derivative, denoted f ''(x) or d²y/dx², is the derivative of the first derivative. It describes the rate of change of the gradient and helps determine concavity.

二阶导数记作 f ''(x) 或 d²y/dx²，是一阶导数的导数。它描述了斜率的变化率，并用于判断曲线的凹凸性。

In kinematics, if s(t) is displacement, then v(t) = s'(t) is velocity and a(t) = s''(t) is acceleration.

在运动学中，若 s(t) 表示位移，则 v(t) = s'(t) 为速度，a(t) = s''(t) 为加速度。

Higher derivatives are found by repeated differentiation. Notation for the n‑th derivative is f⁽ⁿ⁾(x) or dⁿy/dxⁿ.

更高阶的导数通过重复求导得到。n 阶导数记作 f⁽ⁿ⁾(x) 或 dⁿy/dxⁿ。

Examiners often check the ability to find f ''(x) for turning point classification (maximum/minimum/point of inflection).

考官常考查通过 f ''(x) 判定驻点类型（极大值、极小值、拐点）的能力。

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7. Antiderivatives and Indefinite Integrals | 不定积分与原函数

verse process of differentiation. An antiderivative of f(x) is a function F(x) such that F '(x) = f(x).

积分是微分的逆运算。f(x) 的原函数是满足 F '(x) = f(x) 的函数 F(x)。

(x) + C, where C is the constant of integration representing an infinite family of curves.

不定积分写作 ∫ f(x) dx = F(x) + C，其中 C 为积分常数，表示一族曲线。

Never forget the +C in indefinite integrals; omitting it loses marks in both IB and AQA.

不定积分切勿遗漏 +C；在 IB 和 AQA 考试中省略它都会失分。

The fundamental link between differentiation and integration will be exploited in the definite integral section.

微分与积分之间的基本联系将在定积分部分详细展开。

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8. Basic Integration Rules | 基本积分法则

The power rule for integration: ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, valid for n ≠ –1.

幂函数积分法则：∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C，其中 n ≠ –1。

For n = –1, ∫ (1/x) dx = ln |x| + C. Also memorise:

当 n = –1 时，∫ (1/x) dx = ln |x| + C。同时记住：

• ∫ e^x dx = e^x + C   ∫ e^x dx = e^x + C
• ∫ sin x dx = –cos x + C   ∫ sin x dx = –cos x + C
• ∫ cos x dx = sin x + C   ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C   ∫ sec² x dx = tan x + C

Integration is linear, so ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx.

积分具有线性性质：∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx。

Always check your result by differentiation. This is a fast way to confirm the antiderivative is correct.

始终通过求导来检验积分结果。这是确认原函数正确的快捷方法。

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9. Definite Integrals and the Fundamental Theorem | 定积分与微积分基本定理

The definite integral ∫_a^b f(x) dx represents the signed area between the curve y = f(x) and the x‑axis from x = a to x = b.

定积分 ∫_a^b f(x) dx 表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的带符号面积。

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration:

微积分基本定理将求导与积分联系起来：

∫_a^b f(x) dx = F(b) – F(a)   where F ' = f

To evaluate a definite integral, find any antiderivative F, then compute F(b) – F(a). The constant C cancels out.

计算定积分时，先找到一个原函数 F，再求 F(b) – F(a)。积分常数 C 在相减时抵消。

Areas below the x‑axis give a negative contribution. For total area, split the interval where f changes sign.

x 轴下方的部分贡献负值。求总面积时，需在 f 变号的点分割区间并取绝对值。

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10. Integration by Substitution | 换元积分法

ion is the chain rule in reverse. Choose u = g(x) such that the integral simplifies to ∫ f '(u) du.

换元积分法是链式法则的逆用。选取 u = g(x)，使得积分简化为 ∫ f '(u) du 的形式。

nite integrals, remember to change the limits: when x = a, u = g(a); when x = b, u = g(b).

对于定积分，务必变换上下限：当 x = a 时 u = g(a)；当 x = b 时 u = g(b)。

Example: ∫ 2x e^x2 dx. Let u = x², du = 2x dx → ∫ e^u du = e^u + C = e^x2 + C.

示例：∫ 2x e^x2 dx。设 u = x²，则 du = 2x dx → ∫ e^u du = e^u + C = e^x2 + C。

Quadratic denominators often suggest a trigonometric substitution, though AQA and IB focus mainly on polynomial and exponential forms.

具有二次分式的积分常采用三角换元，但 AQA 和 IB 主要考查多项式与指数形式的换元。

Practice identifying the 'inner function' whose derivative is already present in the integrand; this is the key to spotting the right substitution.

练习识别被积函数中已含其导数的那部分“内层函数”；这是看准正确换元的关键。

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