4 Products | 四类乘积运算

📚 4 Products | 四类乘积运算

In AQA A-level Mathematics, the idea of a “product” appears in many different contexts. This article focuses on four essential types of products: algebraic products, the product rule for differentiation, the scalar (dot) product of vectors, and the product rule applied to integration by parts. Mastery of these four products is vital for exam success.

在 AQA A-level 数学中，”乘积”这一概念出现在许多不同的场景里。本文聚焦四种核心乘积类型：代数表达式的乘积、微积分中的乘积法则、向量的数量积（点积），以及由乘积法则引出的分部积分法。掌握这四类乘积对于考试取得好成绩至关重要。

1. Products of Algebraic Expressions | 代数表达式相乘

Expanding products of two or more linear or quadratic expressions is a fundamental skill. For example, (x + 3)(2x – 5) = 2x² + x – 15. You should be able to use the distributive law confidently.

两个或多个一次、二次表达式相乘展开是一项基础技能。例如 (x + 3)(2x – 5) = 2x² + x – 15。你应该能够自信地运用分配律展开。

Key special products include the perfect square and difference of two squares:

常见的特殊乘积包括完全平方和平方差：

(a + b)² = a² + 2ab + b², (a – b)² = a² – 2ab + b², (a + b)(a – b) = a² – b²

These identities are used in factorisation, coordinate geometry, and binomial expansions. For instance, rewriting 9x² – 16 as (3x + 4)(3x – 4) can simplify solving equations or evaluating limits.

这些恒等式用于因式分解、解析几何和二项式展开。例如，将 9x² – 16 改写为 (3x + 4)(3x – 4) 可以简化求解方程或计算极限的过程。

Expand the product of two binomials systematically: multiply each term in the first bracket by each term in the second.
Check for special patterns: perfect squares, difference of squares, and sum/difference of cubes.
When working with more than two factors, expand pairwise and simplify.

系统展开两个二项式的乘积：用第一个括号中的每一项乘以第二个括号中的每一项。
检查特殊模式：完全平方、平方差、立方和/差。
处理两个以上因式时，两两相乘再化简。

2. The Product Rule for Differentiation | 微分中的乘积法则

When a function is the product of two differentiable functions, u(x) and v(x), the derivative is given by the product rule:

当一个函数是两个可导函数 u(x) 与 v(x) 的乘积时，其导数由乘积法则给出：

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

In AQA examinations, you must be able to choose which function to call u and which to call v. It is often convenient to simplify the expression first if possible.

在 AQA 考试中，你必须能够选择哪个函数作为 u、哪个作为 v。如果可以，先化简表达式往往更方便。

Example: Differentiate y = x²·sin x.

例：求 y = x²·sin x 的导数。

Here u = x², v = sin x, so u’ = 2x and v’ = cos x. Hence:

这里 u = x²，v = sin x，所以 u’ = 2x，v’ = cos x。因此：

dy/dx = 2x·sin x + x²·cos x

Remember that the product rule can be extended to more than two functions. For three functions u, v, w:

记住，乘积法则可以推广到两个以上的函数。对于三个函数 u、v、w：

d/dx (uvw) = u’vw + uv’w + uvw’

Do not confuse the product rule with the rule for differentiating a constant multiple.
Always simplify the final derivative where possible, using trigonometric identities if needed.
Check if the expression can be expanded first: for example, y = x(3x² – 2x + 1) is easier to differentiate directly.

不要将乘积法则与常数倍求导法则混淆。
尽可能化简最终导数，必要时使用三角恒等式。
检查表达式是否可以先展开：例如 y = x(3x² – 2x + 1) 直接求导更容易。

3. The Scalar (Dot) Product of Vectors | 向量的数量积（点积）

For two vectors a and b, the scalar product is defined as:

对于两个向量 a 与 b，数量积定义为：

a · b = |a||b|·cos θ

where θ is the angle between the vectors (0 ≤ θ ≤ π). If the vectors are given in component form, a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then:

其中 θ 是两向量之间的夹角（0 ≤ θ ≤ π）。如果向量以分量形式给出，a = (a₁, a₂, a₃)，b = (b₁, b₂, b₃)，则：

a · b = a₁b₁ + a₂b₂ + a₃b₃

The scalar product is a scalar (a number), not a vector. It is used to find the angle between two lines, to check perpendicularity, and in projections.

数量积是一个标量（数），不是向量。它用于求两条直线的夹角、判断垂直以及计算投影。

Important results:

重要结论：

If a · b = 0 and both vectors are non-zero, then a and b are perpendicular.
For parallel vectors, a · b = |a||b| (if they point in the same direction) or -|a||b| (if opposite).
The angle θ in degrees or radians is found using cos θ = (a · b) / (|a||b|).

若 a · b = 0 且两向量均为非零向量，则 a 与 b 垂直。
对于平行向量，若同向则 a · b = |a||b|，若反向则 a · b = -|a||b|。
夹角 θ（度或弧度）由 cos θ = (a · b) / (|a||b|) 求得。

Example: a = (2, -1, 3), b = (4, 0, -1). Then a · b = 2×4 + (-1)×0 + 3×(-1) = 8 – 3 = 5.

例：a = (2, -1, 3)，b = (4, 0, -1)。则 a · b = 2×4 + (-1)×0 + 3×(-1) = 8 – 3 = 5。

4. Products in Integration: Integration by Parts | 积分中的乘积：分部积分法

Integration by parts is derived from the product rule for differentiation. It allows us to integrate products of functions where one part is easily differentiated and the other easily integrated.

分部积分法由微分中的乘积法则推导而来。它用于积分两个函数的乘积，其中一部分容易求导，另一部分容易积分。

The formula is:

公式为：

∫ u dv = uv – ∫ v du

In practice, choose u to be a function that becomes simpler when differentiated (such as x, x², ln x), and choose dv to be a function that is easy to integrate (such as eˣ, cos x, sin x).

在实际操作中，选择 u 为一个求导后更简单的函数（如 x、x²、ln x），选择 dv 为一个容易积分的函数（如 eˣ、cos x、sin x）。

Example: Find ∫ x·eˣ dx.

例：求 ∫ x·eˣ dx。

Let u = x, then du = dx. Let dv = eˣ dx, then v = eˣ. Applying the formula:

令 u = x，则 du = dx。令 dv = eˣ dx，则 v = eˣ。代入公式：

∫ x·eˣ dx = x·eˣ – ∫ eˣ dx = x·eˣ – eˣ + C

For definite integrals, apply the limits to both terms: [x·eˣ]ₐᵇ – ∫ₐᵇ eˣ dx.

对于定积分，对两项分别代入上下限：[x·eˣ]ₐᵇ – ∫ₐᵇ eˣ dx。

Sometimes you need to apply integration by parts more than once, for example with x²·sin x.
When integrating ln x, choose u = ln x and dv = dx.
Remember the constant of integration for indefinite integrals.

有时需要多次使用分部积分，例如 x²·sin x。
积分 ln x 时，令 u = ln x，dv = dx。
不要忘记不定积分中的积分常数。

5. Recognising Products in Exam Questions | 在考试题中识别乘积

AQA examiners often hide products in seemingly complex expressions. In differentiation, look for a function written as the product of two simpler functions. In vectors, determine whether a question requires the scalar product or the angle. In integration, identify whether you can integrate directly or need integration by parts.

AQA 考官经常把乘积隐藏在看似复杂的表达式中。在微分中，观察一个函数是否是两个较简函数之积。在向量中，判断题目要求数量积还是夹角。在积分中，识别是直接积分还是需要分部积分。

Here is a comparison table to help you choose the correct method:

下面是一张比较表，帮助你选择正确的方法：

Situation	Method
Expanding (x + 2)(x – 3)	Algebraic expansion
Differentiating y = x²·cos x	Product rule
Finding angle between two vectors	Scalar product
Integrating ∫ x·ln x dx	Integration by parts

情境	方法
展开 (x + 2)(x – 3)	代数展开
求 y = x²·cos x 的导数	乘积法则
求两向量夹角	数量积
积分 ∫ x·ln x dx	分部积分法

6. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often lose marks in product questions due to simple slips. Here are the most common errors:

学生常在乘积类题目中因简单失误而丢分。以下是最常见的错误：

Writing u’v + uv’ as u’v’ (incorrect!)
Forgetting the minus sign in the difference of two squares: (a – b)(a + b) = a² – b², not a² + b².
Mixing up the scalar product with the vector cross product, which is not in the AQA A-level mathematics specification.
In integration by parts, choosing u and dv the wrong way round, leading to a more complicated integral.

将 u’v + uv’ 写成 u’v’（错误！）
忘记平方差中的减号：(a – b)(a + b) = a² – b²，而不是 a² + b²。
将数量积与向量叉积混淆，AQA A-level 数学大纲不要求叉积。
在分部积分中，将 u 与 dv 选反，导致积分变得更复杂。

Also, always show your working clearly. In AQA, method marks are awarded even if the final answer is wrong, as long as your steps demonstrate the correct product rule or valid substitution.

同时，务必清晰写出步骤。在 AQA 中，只要你的步骤展示出正确的乘积法则或有效的代入，即使最终答案错误，也能获得方法分。

7. Practice Questions | 练习题目

Try these questions to check your understanding of the four products:

尝试以下题目，检查你对这四类乘积的理解：

Expand and simplify (x – 4)².
Differentiate y = (2x + 1)·eˣ.
Vectors a = (3, -2, 1) and b = (1, 4, -2). Find a · b and the angle between them.
Find ∫ x·sin x dx.

展开并化简 (x – 4)²。
求 y = (2x + 1)·eˣ 的导数。
向量 a = (3, -2, 1)，b = (1, 4, -2)。求 a · b 以及它们之间的夹角。
求 ∫ x·sin x dx。

Answers: 1) x² – 8x + 16. 2) dy/dx = 2eˣ + (2x + 1)eˣ = (2x + 3)eˣ. 3) a · b = 3×1 + (-2)×4 + 1×(-2) = -7; |a| = √14, |b| = √21, so cos θ = -7 / (√14 √21). 4) ∫ x·sin x dx = -x·cos x + sin x + C.

答案：1) x² – 8x + 16。2) dy/dx = 2eˣ + (2x + 1)eˣ = (2x + 3)eˣ。3) a · b = 3×1 + (-2)×4 + 1×(-2) = -7；|a| = √14，|b| = √21，所以 cos θ = -7 / (√14 √21)。4) ∫ x·sin x dx = -x·cos x + sin x + C。

8. Conclusion | 总结

The four products — algebraic expansion, the product rule, the scalar product, and integration by parts — form a network of connected ideas in AQA A-level Mathematics. Understanding the relationships between them helps you to apply the correct technique in any context.

这四类乘积——代数展开、乘积法则、数量积和分部积分——在 AQA A-level 数学中构成了相互关联的思想网络。理解它们之间的联系，能帮助你在任何情境中应用正确的技巧。

Keep practising with past papers and focus on clear, methodical working. Products are not just isolated formulas; they are tools that unlock many other topics.

坚持用往年真题练习，并注重清晰、有条理的步骤。乘积不仅仅是孤立的公式，它们是解锁许多其他主题的工具。

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4 Products | 四类乘积运算