Edexcel A-Level Core Pure: Differentiation | Edexcel A-Level 核心纯数学：微分

📚 Edexcel A-Level Core Pure: Differentiation | Edexcel A-Level 核心纯数学：微分

This revision guide covers the essential differentiation techniques required for Edexcel A-Level Mathematics, from first principles to applications such as optimisation and tangents. It is designed as a focused summary for exam preparation, with paired English and Chinese explanations to support bilingual learners.

本复习指南涵盖 Edexcel A-Level 数学所需的微分核心技巧，从第一原理到优化与切线等应用。它旨在为考试复习提供重点总结，并配有中英对照解释，以帮助双语学习者。

1. Introduction to Differentiation | 微分入门

Differentiation measures the rate at which a function changes with respect to its input variable. For a curve y = f(x), the derivative dy/dx gives the gradient of the tangent at any point on the curve.

微分用于衡量函数相对于自变量的变化率。对于曲线 y = f(x)，导数 dy/dx 表示曲线上任意一点处切线的斜率。

dy/dx = lim (Δx → 0) [Δy / Δx]

In Edexcel exam questions, this idea is often tested by asking you to interpret a derivative as a rate of change, a gradient, or an instantaneous speed in a real-world context.

在 Edexcel 考试题中，这一概念常以解释导数作为变化率、斜率或实际问题中的瞬时速度等形式出现。

A key exam tip is to always identify the units of a rate of change when an application question is given: for example, if y is distance in metres and x is time in seconds, dy/dx is measured in metres per second.

一个重要的考试提示是：在应用问题中，一定要识别变化率的单位。例如，若 y 表示以米为单位的距离，x 表示以秒为单位的时间，则 dy/dx 的单位是米每秒。

2. Differentiation from First Principles | 第一原理微分

The formal definition of the derivative uses a limit. For a function f(x), the derivative f'(x) is given by the following expression.

导数的正式定义使用极限。对于函数 f(x)，导数 f'(x) 由以下表达式给出。

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

For example, to differentiate f(x) = x² from first principles, expand (x + h)² – x² = 2xh + h², divide by h, then let h tend to 0 to obtain 2x. This process is a common proof question in Edexcel Pure Mathematics.

例如，用第一原理求 f(x) = x² 的导数时，先展开 (x + h)² – x² = 2xh + h²，再除以 h，然后令 h 趋近于 0，得到 2x。这个过程是 Edexcel 纯数学中常见的证明题。

When using first principles, do not substitute h = 0 before simplifying the fraction. The limit must be evaluated after cancelling any common factor of h in the numerator and denominator.

使用第一原理时，不要在化简分式之前代入 h = 0。必须在分子和分母中消去公因子 h 之后再计算极限。

3. Power, Constant Multiple, Sum and Difference Rules | 幂法则、常数倍法则、和差法则

The power rule is the most frequently used differentiation rule in A-Level maths. If f(x) = xⁿ, where n is any real constant, then f'(x) = n xⁿ⁻¹.

幂法则是 A-Level 数学中最常用的微分法则。如果 f(x) = xⁿ，其中 n 为任意实常数，则 f'(x) = n xⁿ⁻¹。

d/dx (xⁿ) = n xⁿ⁻¹

You can differentiate term by term. For example, if y = 5x³ – 2x + 7, then dy/dx = 15x² – 2. Constant multiples stay unchanged, and the derivative of a constant is zero.

你可以逐项求导。例如，若 y = 5x³ – 2x + 7，则 dy/dx = 15x² – 2。常数倍保持不变，常数的导数为零。

It is often useful to rewrite expressions using negative and fractional indices before differentiating: for example, 1/x² = x⁻² and √x = x^½. This allows the power rule to be applied directly.

在求导前，通常需要先将表达式改写为负指数和分数指数形式：例如，1/x² = x⁻²，√x = x^½。这样就可以直接使用幂法则。

Common errors include forgetting to reduce the power by 1 or failing to multiply by the original power. Always check your final answer by mentally differentiating back.

常见错误包括忘记将指数减 1，或忘记乘以原来的指数。始终通过心算重新求导来检查最终答案是否正确。

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

When differentiating the product of two functions u(x) and v(x), use the product rule: dy/dx = u’v + uv’. Here u’ means du/dx and v’ means dv/dx.

在对两个函数 u(x) 和 v(x) 的乘积求导时，使用乘积法则：dy/dx = u’v + uv’。这里 u’ 表示 du/dx，v’ 表示 dv/dx。

d/dx (uv) = u(dv/dx) + v(du/dx)

The quotient rule applies when one function is divided by another. If y = u/v, then dy/dx = (v u’ – u v’) / v². It is important to keep the order in the numerator correct.

当一个函数除以另一个函数时，使用商法则。若 y = u/v，则

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