UEC 数学宝典 – 独中统考数学讲义 SC04 SC05 SC06 SC07

📌 SC07 高级数学 II — 统考最难科目，本宝典优先覆盖。

📌 SC07 Advanced Mathematics II — hardest UEC subject. Notes prioritised here.

章节列表

Unit List

第1章 · 微分

Unit 1 · Differentiation

Differentiation

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第2章 · 积分

Unit 2 · Integration

Integration

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第3章 · 数列与级数

Unit 3 · Sequences & Series

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第4章 · 向量

Unit 4 · Vectors

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第5章 · 复数

Unit 5 · Complex Numbers

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SC07 · UNIT 1

微分

Differentiation

难度：★★★★☆ · 预备知识：代数、极限基础

Difficulty: ★★★★☆ · Prerequisites: Algebra, Basic Limits

📌 核心概念

📌 Core Concepts

微分（Differentiation）是求函数变化率的运算。导数 $f'(x)$ 表示函数 $f(x)$ 在 $x$ 处的瞬时变化率，几何上即切线的斜率。

Differentiation finds the rate of change of a function. The derivative $f'(x)$ represents the instantaneous rate of change of $f(x)$ at $x$, which geometrically is the slope of the tangent line.

📋 关键公式

📋 Key Formulas

基本求导法则

Basic Differentiation Rules

幂次法则：$$\frac{d}{dx}[x^n] = nx^{n-1}$$

乘积法则：$$\frac{d}{dx}[uv] = u'v + uv'$$

商法则：$$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$$

链式法则：$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

常用函数导数

Standard Derivatives

$\frac{d}{dx}[\sin x] = \cos x$

$\frac{d}{dx}[\cos x] = -\sin x$

$\frac{d}{dx}[\tan x] = \sec^2 x$

$\frac{d}{dx}[e^x] = e^x$

$\frac{d}{dx}[\ln x] = \dfrac{1}{x}$

$\frac{d}{dx}[a^x] = a^x \ln a$

✏️ 例题详解

✏️ Worked Examples

例题 1 — 链式法则应用

Example 1 — Chain Rule

求 $y = \sin(3x^2 + 1)$ 的导数。

Find the derivative of $y = \sin(3x^2 + 1)$.

设 $u = 3x^2 + 1$，则 $y = \sin u$。

Let $u = 3x^2 + 1$, so $y = \sin u$.

求各部分导数：$\dfrac{dy}{du} = \cos u$，$\dfrac{du}{dx} = 6x$

Find each part: $\dfrac{dy}{du} = \cos u$, $\dfrac{du}{dx} = 6x$

代入链式法则：$$\frac{dy}{dx} = \cos u \cdot 6x = 6x\cos(3x^2+1)$$

Apply chain rule: $$\frac{dy}{dx} = \cos u \cdot 6x = 6x\cos(3x^2+1)$$

例题 2 — 乘积法则

Example 2 — Product Rule

求 $y = x^2 e^x$ 的导数。

Find the derivative of $y = x^2 e^x$.

设 $u = x^2$，$v = e^x$，则 $u' = 2x$，$v' = e^x$。

Let $u = x^2$, $v = e^x$, so $u' = 2x$, $v' = e^x$.

代入乘积法则：$$\frac{dy}{dx} = u'v + uv' = 2xe^x + x^2e^x = xe^x(2+x)$$

Apply product rule: $$\frac{dy}{dx} = u'v + uv' = 2xe^x + x^2e^x = xe^x(2+x)$$

⚠️ 常见错误

⚠️ Common Mistakes

❌ 忘记链式法则：对复合函数直接求导，漏乘内层导数。如 $\frac{d}{dx}[\sin(3x)] = \cos(3x)$（少乘了 3）。
❌ Forgetting chain rule: Differentiating composite functions without multiplying by the inner derivative. E.g. writing $\frac{d}{dx}[\sin(3x)] = \cos(3x)$ instead of $3\cos(3x)$.
❌ 乘积法则混淆：误用 $(uv)' = u'v'$，正确应为 $u'v + uv'$。
❌ Wrong product rule: Using $(uv)' = u'v'$ instead of $u'v + uv'$.
❌ $\ln x$ 导数写错：$\frac{d}{dx}[\ln x] = \frac{1}{x}$，而非 $\frac{1}{\ln x}$。
❌ Wrong $\ln x$ derivative: $\frac{d}{dx}[\ln x] = \frac{1}{x}$, not $\frac{1}{\ln x}$.

❓ 常见问题

❓ FAQ

隐函数微分和显函数微分有什么区别？

What is the difference between implicit and explicit differentiation?

显函数可以直接写成 $y = f(x)$ 的形式，直接求导即可。隐函数（如 $x^2 + y^2 = 1$）无法直接分离，需对两边同时对 $x$ 求导，并利用链式法则处理含 $y$ 的项（每次出现 $y$ 都要乘 $\frac{dy}{dx}$）。

An explicit function can be written as $y = f(x)$ and differentiated directly. An implicit function (e.g. $x^2 + y^2 = 1$) cannot be isolated, so you differentiate both sides with respect to $x$, applying the chain rule to any $y$ terms (multiplying by $\frac{dy}{dx}$ each time $y$ appears).

二阶导数是什么？有什么用？

What is a second derivative and what is it used for?

二阶导数 $f''(x)$ 是对导数再求一次导。它描述函数的凹凸性：$f''(x) > 0$ 时函数下凸（向上弯），$f''(x) < 0$ 时上凸（向下弯）。在统考中常用于判断极值是极大还是极小。

The second derivative $f''(x)$ is the derivative of the derivative. It describes the concavity of the function: $f''(x) > 0$ means concave up, $f''(x) < 0$ means concave down. In UEC exams it is commonly used to classify stationary points as maxima or minima.

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UEC 数学宝典 UEC Math Wiki

章节列表

Unit List

微分

Differentiation

📌 核心概念

📌 Core Concepts

📋 关键公式

📋 Key Formulas

✏️ 例题详解

✏️ Worked Examples

⚠️ 常见错误

⚠️ Common Mistakes

❓ 常见问题

❓ FAQ