The Product Rule | 乘积法则

📚 The Product Rule | 乘积法则

In A-Level Mathematics, differentiation is a core skill. The product rule allows you to differentiate expressions made by multiplying two functions together, such as x² sin x or (3x+1)e^x. This topic appears regularly in Edexcel Pure Mathematics exam papers and is essential for rates of change, optimisation, and integration by parts later.

在A-Level数学中，微分是一项核心技能。乘积法则用于对两个函数相乘而成的表达式求导，例如 x² sin x 或 (3x+1)e^x。该内容在Edexcel纯数学试卷中经常出现，对后续变化率、最优化以及分部积分法至关重要。

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1. What Is the Product Rule? | 什么是乘积法则

The product rule is a differentiation rule used when you need to differentiate a function that is the product of two simpler functions. If y = u(x)v(x), the derivative is not simply u'(x)v'(x). Instead, the rule combines both functions and both derivatives in a specific way.

乘积法则是一种微分规则，用于对两个较简单函数相乘得到的函数求导。如果 y = u(x)v(x)，其导数并不是简单的 u'(x)v'(x)。该法则以特定方式同时结合两个函数和它们的导数。

In other words, when two changing quantities multiply together, their combined rate of change depends on both factors. That is why the product rule has two separate terms.

换句话说，当两个变化的量相乘时，它们的总变化率取决于两个因子。这就是乘积法则包含两个独立项的原因。

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2. The Formula and Notation | 公式与记号

For y = uv, where u and v are functions of x, the product rule states:

设 y = uv，其中 u 和 v 都是 x 的函数，则乘积法则为：

dy/dx = u · dv/dx + v · du/dx

Alternatively, using prime notation, if y = uv then y’ = u v’ + v u’. It is common to write f'(x) = u(x)v'(x) + v(x)u'(x).

也可以使用撇号记法：若 y = uv，则 y’ = u v’ + v u’。通常也写成 f'(x) = u(x)v'(x) + v(x)u'(x)。

You may see the rule written as “first times derivative of second plus second times derivative of first”. This memory aid is useful under exam pressure.

你可能会看到该法则写作“第一个函数乘第二个函数的导数，加上第二个函数乘第一个函数的导数”。这个记忆口诀在考试压力下非常实用。

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3. Why the Product Rule Is Needed | 为什么需要乘积法则

A tempting mistake is to assume that the derivative of a product equals the product of the derivatives. For example, (x² sin x)’ is not 2x cos x. You can verify this by testing a simple case: if y = x · x = x², the product rule gives dy/dx = x(1) + x(1) = 2x, which matches the power rule; the false rule would give 1 × 1 = 1, which is wrong.

一个常见的错误是假设乘积的导数等于导数的乘积。例如 (x² sin x)’ 并不是 2x cos x。你可以用一个简单例子验证：若 y = x · x = x²，乘积法则给出 dy/dx = x(1) + x(1) = 2x，与幂法则一致；而错误的法则会得到 1 × 1 = 1，这是错误的。

The product rule captures the interaction between the two changing factors. Because both u and v vary with x, the total rate of change contains two contributions.

乘积法则体现的是两个变化因子之间的相互作用。由于 u 和 v 都随 x 变化，总变化率包含两个贡献部分。

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4. A Basic Worked Example | 基础例题

Differentiate y = x² sin x with respect to x.

对 y = x² sin x 关于 x 求导。

Let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x. Substitute into the product rule:

设 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x。代入乘积法则：

dy/dx = x² · cos x + sin x · 2x

So dy/dx = x² cos x + 2x sin x. It is acceptable to leave the answer in factorised or expanded form unless the question requests a specific form.

因此 dy/dx = x² cos x + 2x sin x。除非题目要求特定形式，答案可以保留因式分解或展开形式。

Notice that the two terms are not like terms, so they cannot be further combined. Always re-read the question to see if simplification is required.

注意这两项不是同类项，因此不能进一步合并。务必重新审题，看是否需要化简。

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5. Choosing u and v Smartly | 合理选择 u 和 v

Although the product rule is symmetric and works whichever way you label the factors, a smart choice can reduce sign errors. Often you should let u be the polynomial or simpler factor and v be the trigonometric, exponential, or logarithmic factor. For example, in y = x³ e^x, choose u = x³ and v = e^x.

虽然乘积法则是对称的，无论你如何标记两个因子都成立，但合理选择可以减少符号错误。通常可以让 u 为多项式或较简单的因子，让 v 为三角函数、指数函数或对数函数因子。例如在 y = x³ e^x 中，选择 u = x³，v = e^x。

This choice does not change the final answer, but it often makes the written derivative clearer. Keep u and v clearly labelled before differentiating to avoid mixing up the factors.

这种选择不会改变最终答案，但通常会使书写更清晰。求导前清晰地标出 u 和 v，避免混淆因子。

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6. Combining Product Rule with Chain Rule | 乘积法则与链式法则结合

Many Edexcel exam questions require both the product rule and the chain rule in the same derivative. This happens when one factor is a composite function, such as (2x+1)³ or ln(3x-1). Differentiate the composite factor using the chain rule first, then apply the product rule.

许多Edexcel考题要求在同一次求导中同时使用乘积法则和链式法则。当一个因子是复合函数时（例如 (2x+1)³ 或 ln(3x-1)）就会出现这种情况。先用链式法则对复合因子求导，再应用乘积法则。

Example: Differentiate y = x²(2x+1)³.

例题：对 y = x²(2x+1)³ 求导。

Let u = x² and v = (2x+1)³. Then du/dx = 2x. For v, use the chain rule: dv/dx = 3(2x+1)² × 2 = 6(2x+1)².

设 u = x²，v = (2x+1)³。则 du/dx = 2x。对 v 使用链式法则：dv/dx = 3(2x+1)² × 2 = 6(2x+1)²。

Applying the product rule gives:

应用乘积法则得到：

dy/dx = x² · 6(2x+1)² + (2x+1)³ · 2x

This can be factorised as 2x(2x+1)² [3x + (2x+1)] = 2x(2x+1)²(5x+1). In an exam, factorisation often

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