A-Level Edexcel Mathematics: Core Techniques in Differentiation and Integration | A-Level Edexcel 数学：微分与积分核心技巧

📚 A-Level Edexcel Mathematics: Core Techniques in Differentiation and Integration | A-Level Edexcel 数学：微分与积分核心技巧

Welcome to this revision guide for the Edexcel A-Level Mathematics Pure specification. Differentiation and integration are two of the most heavily examined topics across Paper 1 and Paper 2. This article focuses on the core rules, standard results, and key applications you need to master for high marks.

欢迎阅读本复习指南，针对 Edexcel A-Level 数学纯数部分。微分与积分是第一卷和第二卷中考查频率最高的两个主题。本文聚焦于核心法则、标准结论以及必须掌握的关键应用，帮助你在考试中取得高分。

---

1. The Gradient Function and First Principles | 导数与梯度函数

The derivative of a function f(x) gives the gradient of the tangent at any point. It is defined by the first-principles limit:

函数 f(x) 的导数给出任意点处切线的斜率。它由第一原理的极限定义。

f ‘(x) = lim(h→0) [f(x+h) – f(x)] / h

For example, when f(x) = x², expanding f(x+h) = (x+h)² = x² + 2xh + h² leads to the limit of 2x + h, so the derivative is 2x.

例如，当 f(x) = x² 时，展开 f(x+h) = (x+h)² = x² + 2xh + h²，极限为 2x + h，因此导数为 2x。

In Edexcel papers, first-principles questions appear regularly in the Pure 1 paper, so you must be confident with the expansion and limiting process.

在 Edexcel 试卷中，第一原理的题目经常出现在 Pure 1 卷中，因此你必须熟练掌握展开与极限过程。

---

2. The Power Rule for Differentiation | 幂函数微分法则

For any real power n, the power rule states:

对于任意实数次幂 n，幂函数法则为：

d/dx (xⁿ) = n xⁿ⁻¹

This rule also works with rational and negative powers. For instance, d/dx (√x) = d/dx (x^½) = (1/2)x^-½ = 1/(2√x).

该法则同样适用于有理数和负数次幂。例如，d/dx (√x) = d/dx (x^½) = (1/2)x^-½ = 1/(2√x)。

Edexcel often tests this by asking you to rewrite terms such as 1/x³ as x⁻³ before differentiating. You should also be able to use the rule for terms like 5x⁴ – 2x² + 7x – 9, where the derivative is 20x³ – 4x + 7.

Edexcel 常考先将 1/x³ 改写为 x⁻³ 再求导。你还应能对 5x⁴ – 2x² + 7x – 9 这类多项式使用该法则，其导数为 20x³ – 4x + 7。

---

3. The Chain Rule | 链式法则

To differentiate a composite function y = f(g(x)), use the chain rule:

对复合函数 y = f(g(x)) 求导，使用链式法则：

dy/dx = dy/du × du/dx

Example: if y = (3x + 5)⁴, let u = 3x + 5. Then y = u⁴, dy/du = 4u³, du/dx = 3. Therefore dy/dx = 12(3x + 5)³.

示例：若 y = (3x + 5)⁴，设 u = 3x + 5。则 y = u⁴，dy/du = 4u³，du/dx = 3。因此 dy/dx = 12(3x + 5)³。

In A-Level papers, the chain rule is commonly combined with exponential, logarithmic and trigonometric functions. For example, the derivative of e^(2x² – 3x) is (4x – 3)e^(2x² – 3x).

在 A-Level 试卷中，链式法则常与指数、对数和三角函数结合考查。例如，e^(2x² – 3x) 的导数为 (4x – 3)e^(2x² – 3x)。

---

4. The Product Rule | 乘积法则

When y = u(x)v(x), the derivative is:

当 y = u(x)v(x) 时，导数为：

dy/dx = u’v + uv’

Example: for y = x² sin x, let u = x² and v = sin x. Then u’ = 2x, v’ = cos x, so dy/dx = 2x sin x + x² cos x.

示例：对于 y = x² sin x，设 u = x²，v = sin x。则 u’ = 2x，v’ = cos x，所以 dy/dx = 2x sin x + x² cos x。

Always show your choice of u and v clearly in working to avoid sign or order errors. If a product has more than two factors, apply the rule repeatedly or use the logarithmic method if required.

解题时应清晰写出 u 和 v 的选择，以避免符号或顺序错误。如果乘积包含两个以上因式，可重复使用乘积法则，或按题目要求使用对数求导法。

---

5. The Quotient Rule | 商法则

For y = u/v, the quotient rule is:

对于 y = u/v，商法则为：

dy/dx = (u’v – uv’) / v²

Example: if y = x / (x² + 1), then u = x, v = x² + 1. Hence u’ = 1, v’ = 2x, and dy/dx = [(1)(x²+1) – (x)(2x)] / (x²+1)² = (1 – x²) / (x²+1)².

示例：若 y = x / (x² + 1)，则 u = x，v = x² + 1。因此 u’ = 1，v’ = 2x，dy/dx = [(1)(x²+1) – (x)(2x)] / (x²+1)² = (1 – x²) / (x²+1)²。

The quotient rule is most useful when the numerator and denominator are both non-trivial functions of x. If the denominator is a constant, simply divide or use the chain rule instead.

当分子和分母都是非平凡的函数时，商法则最为有用。如果分母是常数，直接拆项或使用链式法则即可。

---

6. Differentiating Standard Functions | 标准函数微分

You must memorise these standard derivatives for Edexcel:

你必须熟记以下 Edexcel 标准导数：

• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec² x
• d/dx (eˣ) = eˣ
• d/dx (ln x) = 1/x

When these functions have a linear argument, such as sin(ax + b), multiply by the coefficient of x. For example, d/dx [sin(5x – 2)] = 5 cos(5x – 2).

当这些函数的自变量为线性形式如 sin(ax + b) 时，需要乘以 x 的系数。例如，d/dx [sin(5x – 2)] = 5 cos(5x – 2)。

For composite arguments, combine the standard derivative with the chain rule. Thus d/dx [ln(3x + 1)] = 3 / (3x + 1).

对于复合自变量，将标准导数与链式法则结合使用。因此 d/dx [ln(3x + 1)] = 3 / (3x + 1)。

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）