Calculus Essentials: Differentiation and Integration | 微积分核心：微分与积分

📚 Calculus Essentials: Differentiation and Integration | 微积分核心：微分与积分

Calculus is one of the most important branches of A-level Mathematics. It provides the tools to describe change and accumulation, forming the foundation for advanced study in physics, engineering, and economics. This revision guide covers the key techniques of differentiation and integration, including rules, applications, and common pitfalls. Each section is written in a bilingual format to support both understanding and exam preparation.

微积分是 A-level 数学最重要的分支之一。它提供了描述变化和累积的工具，为物理、工程和经济等学科的高阶学习奠定了基础。本复习指南涵盖微分与积分的关键技巧，包括法则、应用和常见易错点。每个小节均采用双语形式，帮助理解并备考。

1. Understanding Differentiation | 理解微分

angent to the curve at that point. For a function y = f(x), the derivative is written as f'(x) or dy/dx.

微分衡量函数相对于自变量的变化速率。从几何上看，函数在某点的导数就是曲线在该点切线的斜率。对于函数 y = f(x)，导数记作 f'(x) 或 dy/dx。

The formal definition uses the limit of the difference quotient. In practice, most A-level questions require applying standard rules rather than deriving from first principles, but understanding the limit idea helps when dealing with unusual functions.

正式定义使用差商的极限。在实际考试中，大多数 A-level 题目要求应用标准法则，而不需要从第一原理推导，但理解极限思想有助于处理特殊函数。

In kinematics, differentiation links displacement, velocity, and acceleration. If s is displacement and t is time, then ds/dt is velocity and dv/dt is acceleration. This interpretation appears in mechanics and applied mathematics questions.

在运动学中，微分将位移、速度和加速度联系起来。若 s 为位移、t 为时间，则 ds/dt 为速度，dv/dt 为加速度。这一解释常出现在力学和应用数学题中。

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2. The Power Rule and Basic Derivatives | 幂函数法则与基本导数

The power rule is the foundation of differentiation. For any real constant n, the derivative of xⁿ is given by:

幂函数法则是微分的基础。对于任意实常数 n，xⁿ 的导数为：

d/dx (xⁿ) = n xⁿ⁻¹

This rule applies to positive integers, negative exponents, and fractional exponents. For example, the derivative of x³ is 3x², the derivative of x⁻² is -2x⁻³, and the derivative of x¹ᐟ² is ½ x⁻¹ᐟ².

该法则适用于正整数指数、负指数和分数指数。例如，x³ 的导数是 3x²，x⁻² 的导数是 -2x⁻³，x¹ᐟ² 的导数是 ½ x⁻¹ᐟ²。

Constants have derivative zero because they do not change. The derivative of a constant multiple c·f(x) is c·f'(x), and the derivative of a sum is the sum of the derivatives. These linearity properties allow us to differentiate term by term.

常数项导数为零，因为常数不发生变化。常数倍 c·f(x) 的导数是 c·f'(x)，和函数的导数是各部分导数之和。这些线性性质允许我们逐项求导。

term to obtain dy/dx = 20x⁴ - 6x. The constant term 7 contributes zero to the derivative.

对于多项式 y = 4x⁵ - 3x² + 7，逐项求导得 dy/dx = 20x⁴ - 6x。常数项 7 对导数的贡献为零。

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3. The Chain Rule for Composite Functions | 复合函数链式法则

r more functions. If y = f(u) and u = g(x), then the derivative of y with respect to x is:

链式法则用于复合函数求导。若 y = f(u) 且 u = g(x)，则 y 对 x 的导数为：

dy/dx = dy/du × du/dx

, dy/du = 4u³, and du/dx = 6x. Multiplying gives dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.

例如，求 y = (3x² + 5)⁴ 的导数，设 u = 3x² + 5。则 y = u⁴，dy/du = 4u³，du/dx = 6x。相乘得 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。

ential and trigonometric composites, the chain rule gives standard results such as d/dx eᵏˣ = k eᵏˣ and d/dx sin(kx) = k cos(kx). Always multiply by the derivative of the inner function.

对于指数和三角复合函数，链式法则给出标准结果，如 d/dx eᵏˣ = k eᵏˣ 和 d/dx sin(kx) = k cos(kx)。务必乘以内层函数的导数。

In harder problems, the chain rule may need to be applied more than once, especially with nested functions. Always identify the outside function first and then the inside function to avoid missing factors.

在较难的问题中，链式法则可能需多次使用，特别是嵌套函数。务必先识别外层函数，再识别内层函数，以免遗漏因子。

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

When two functions are multiplied, the product rule states:

当两个函数相乘时，乘积法则为：

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

For division, the quotient rule states:

对于相除，商法则为：

d/dx (u/v) = (u' v - u v') / v²

Here u and v are functions of x, and u' and v' are their derivatives. The product rule is often remembered as "the first times the derivative of the second plus the second times the derivative of the first".

这里 u 和 v 是 x 的函数，u' 和 v' 是它们的导数。乘积法则常记为“前导后不导加后导前不导”。

The quotient rule should be used when the denominator is not constant. Always keep v² positive in the denominator. A common error is reversing the subtraction in the numerator, so double-check your signs.

当分母不是常数时应使用商法则。分母中的 v² 始终为正。分子中减法顺序是常见易错点，务必检查符号。

If the denominator is a simple monomial, it is often easier to rewrite the expression using negative exponents and apply the power rule with the chain rule instead of the quotient rule. This can save time in exams.Published by TutorHao | Mathematics Revision Series | aleveler.com

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