A-Level Edexcel Maths: Essential Differentiation Techniques | A-Level Edexcel 数学：核心微分方法

📚 A-Level Edexcel Maths: Essential Differentiation Techniques | A-Level Edexcel 数学：核心微分方法

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It underpins problems in pure mathematics, mechanics, and statistics, and appears in both AS and A2 papers. This revision guide covers the essential techniques and common pitfalls to help you master differentiation efficiently.

微分是 Edexcel A-Level 数学中考查最频繁的主题之一。它是纯数学、力学和统计问题的基础，并出现在 AS 和 A2 试卷中。本复习指南涵盖了核心方法与常见易错点，帮助你高效掌握微分。

1. The Derivative and First Principles | 导数与第一原理

The derivative measures the instantaneous rate of change of a function. For a curve y = f(x), the derivative dy/dx is defined as the limit of the average rate of change as the interval tends to zero.

导数衡量函数的瞬时变化率。对于曲线 y = f(x)，导数 dy/dx 定义为当区间趋于零时平均变化率的极限。

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

In Edexcel exams, you may be asked to differentiate a simple function such as x² or x³ from first principles. This tests your understanding of the limit definition rather than just the rule.

在 Edexcel 考试中，你可能会被要求从第一原理求 x² 或 x³ 等简单函数的导数。这考查你对极限定义的理解，而不仅仅是套用公式。

For example, if f(x) = x², then f(x + h) = x² + 2xh + h². The difference quotient simplifies to 2x + h, and letting h → 0 gives dy/dx = 2x.

例如，若 f(x) = x²，则 f(x + h) = x² + 2xh + h²。差分商化简为 2x + h，令 h → 0，得到 dy/dx = 2x。

2. Power Rule and Basic Derivatives | 幂法则与基本导数

The power rule states that the derivative of xⁿ is n xⁿ⁻¹ for any real constant n. It works for positive integers, negative integers, and fractional powers, so it is a versatile tool.

幂法则表明，xⁿ 的导数为 n xⁿ⁻¹，其中 n 为任意实常数。它适用于正整数、负整数和分数次幂，因此用途广泛。

d/dx (xⁿ) = n xⁿ⁻¹

• d/dx (x⁵) = 5x⁴
• d/dx (1/x) = d/dx (x⁻¹) = -x⁻²
• d/dx (√x) = 1/(2√x)

Remember to rewrite rational and negative powers before differentiating. For example, 1/x² becomes x⁻² and 1/√x becomes x⁻½ before applying the rule.

在求导前，记得将有理次数和负次数改写成幂的形式。例如，1/x² 改写为 x⁻²，1/√x 改写为 x⁻½，然后再应用法则。

Linear combinations are differentiated term by term. The derivative of 4x³ – 5x + 7 is 12x² – 5, because constants vanish and the derivative of x is 1.

线性组合可以逐项求导。4x³ – 5x + 7 的导数为 12x² – 5，因为常数项导数为零，x 的导数为 1。

3. Chain Rule | 链式法则

The chain rule is used to differentiate composite functions, such as (3x + 2)⁵ or sin(2x). It is often the most frequently applied rule in A-Level exam questions.

链式法则用于对复合函数求导，例如 (3x + 2)⁵ 或 sin(2x)。它是 A-Level 考题中最常用的法则之一。

Let y = f(u) and u = g(x). Then dy/dx = dy/du × du/dx. In practice, you differentiate the outer function and multiply by the derivative of the inner function.

设 y = f(u)，u = g(x)，则 dy/dx = dy/du × du/dx。实际操作中，先对外层函数求导，再乘以内层函数的导数。

dy/dx = dy/du × du/dx

For y = (3x + 2)⁵, let u = 3x + 2, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 3, giving dy/dx = 15(3x + 2)⁴.

对于 y = (3x + 2)⁵，令 u = 3x + 2，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 3，故 dy/dx = 15(3x + 2)⁴。

Always identify the inner function clearly. In y = e²ˣ, the inner function is 2x, so the derivative is 2e²ˣ.

要清楚地识别内层函数。在 y = e²ˣ 中，内层函数是 2x，因此导数为 2e²ˣ。

4. Product Rule | 乘积法则

When differentiating a product of two functions y = u(x)v(x), use the product rule. It is commonly tested with functions such as x²eˣ or x ln x.

当对两个函数的乘积 y = u(x)v(x) 求导

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