The Product Rule | 乘积法则（微分法）

📚 The Product Rule | 乘积法则（微分法）

In A-Level Mathematics, the product rule is an essential differentiation technique. It lets you find the derivative of a function that is written as the product of two (or more) simpler functions.

在 A-Level 数学中，乘积法则是重要的微分技巧。它帮助我们求两个（或多个）较简单函数相乘所得函数的导数。

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1. What Is a 'Product' in Calculus? | 微积分中的“乘积”是什么？

A product is simply the result of multiplying two or more expressions together. For example, in the expression f(x) = x² sin x, the function f is the product of x² and sin x.

乘积就是两个或多个表达式相乘的结果。例如，在 f(x) = x² sin x 中，函数 f 是 x² 与 sin x 的乘积。

When you try to differentiate such a product, you cannot just differentiate each factor separately and multiply them. You must use the product rule.

当你要对这样的乘积求导时，不能简单地分别对每个因子求导再相乘，而必须使用乘积法则。

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2. The Product Rule Formula | 乘积法则公式

Suppose u(x) and v(x) are differentiable functions. Then the derivative of their product is given by:

设 u(x) 和 v(x) 是可导函数，则它们乘积的导数由下式给出：

ddx[u v] = u dvdx + v dudx

In words: "first times the derivative of the second, plus second times the derivative of the first."

用语言表述就是：“第一个乘上第二个的导数，加上第二个乘上第一个的导数。”

You may also see this written as ddx(uv) = u v' + v u'. Both forms are identical.

你也可能看到写作 ddx(uv) = u v' + v u'，这两种形式完全相同。

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3. Where Does the Product Rule Come From? | 乘积法则从何而来？

The product rule can be derived from first principles using the definition of the derivative. Consider a small change Δ x, leading to changes Δ u and Δ v.

乘积法则可以从导数的定义出发，利用极限原理推导。考虑一个微小的变化量 Δ x，相应地引起 Δ u 和 Δ v。

Then Δ(uv) = (u+Δ u)(v+Δ v) - uv = uΔ v + vΔ u + Δ u Δ v. Dividing by Δ x and letting Δ x → 0 gives the formula.

于是 Δ(uv) = (u+Δ u)(v+Δ v) - uv = uΔ v + vΔ u + Δ u Δ v。除以 Δ x 并令 Δ x → 0，就得到公式。

The term Δ u Δ v disappears because it is negligible compared with the linear terms as Δ x → 0.

其中 Δ u Δ v 这一项在 Δ x → 0 时与线性项相比可以忽略不计。

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4. Worked Example 1: Polynomials | 例 1：多项式

Differentiate y = (3x² + 1)(2x - 5).

求 y = (3x² + 1)(2x - 5) 的导数。

Let u = 3x² + 1 and v = 2x - 5. Then dudx = 6x and dvdx = 2.

令 u = 3x² + 1，v = 2x - 5。则 dudx = 6x，dvdx = 2。

Using the product rule:

应用乘积法则：

dydx = (3x² + 1)(2) + (2x - 5)(6x)

Simplify: dydx = 6x² + 2 + 12x² - 30x = 18x² - 30x + 2.

化简：dydx = 6x² + 2 + 12x² - 30x = 18x² - 30x + 2。

Notice that expanding the original expression first would give the same answer, but the product rule is often faster, especially for higher powers.

注意，先把原式展开再求导也能得到相同结果，但乘积法则通常更快，尤其对于高次幂。

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5. Worked Example 2: Trigonometry | 例 2：三角函数

Differentiate y = x² sin x.

求 y = x² sin x 的导数。

Let u = x² and v = sin x. Then dudx = 2x and dvdx = cos x.

令 u = x²，v = sin x。则 dudx = 2x，dvdx = cos x。

Apply the product rule:

应用乘积法则：

dydx = x² cos x + 2x sin x

This can be written as x(xcos x + 2sin x).

可以写成 x(xcos x + 2sin x)。

This example shows that the product rule is necessary because differentiation is not multiplicative.

这个例子说明乘积法则的必要性，因为导数并不具备乘法性质。

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6. Combined with the Chain Rule | 与链式法则结合

Often a product involves a composite function. For example, y = x e^3x requires both the product rule and the chain rule.

很多乘积中包含复合函数。例如，y = x e^3x 需要同时使用乘积法则和链式法则。

Let u = x and v = e^3x. Then dudx = 1, and to find dvdx we use the chain rule: ddx e^3x = 3e^3x.

令 u = x，v = e^3x。则 dudx = 1，而求 dvdx 需要用到链式法则：ddx e^3x = 3e^3x。

So:

因此：

dydx = x · 3e^3x + e^3x · 1 = e^3x(3x + 1)

Always identify both rules before differentiating. The product rule handles the multiplication, while the chain rule handles the inner function.

求导前务必先识别两个法则。乘积法则处理乘法，链式法则处理内层函数。

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7. Products with Logarithms | 含对数的乘积

Consider y = ln x · cos x. Here u = ln x and v = cos x.

考虑 y = ln x · cos x。这里 u = ln x，v = cos x。

We have dudx = 1x and dvdx = -sin x.

我们有 dudx = 1x，dvdx = -sin x。

Thus:

因此：

dydx = ln x (-sin x) + cos x (1x) = cos xx - sin x ln x

Notice that the order of the two terms does not matter, but it is often sensible to write the term with ln x second to keep signs clear.

注意两个项的先后顺序无关，但为了符号清晰，通常把含 ln x 的项写在后面。

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8. Product Rule with Three Factors | 三个因子的乘积法则

When you have a product of three functions, you can apply the product rule twice. For y = u v w,

当三个函数相乘时，可以分两次使用乘积法则。对于 y = u v w，

dydx = u v w' + u w v' + v w u'

This is a pattern: differentiate one factor at a time while keeping the others unchanged, then add.

规律是：每次只对一个因子求导，保持其他因子不变，然后相加。

For example, y = x² sin x cos x. Differentiate each factor:

例如，y = x² sin x cos x。对每个因子分别求导：

dydx = x² sin x (-sin x) + x² cos x (cos x) + 2x sin x cos x

Then simplify using trigonometric identities if required.

然后根据需要利用三角恒等式进行化简。

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

• Mistake: Writing dydx = u' v' instead of u v' + v u'.
• 错误：写成 dydx = u' v'，而不是 u v' + v u'。

• Mistake: Forgetting to apply the chain rule to one of the factors when it is a composite function.
• 错误：当某个因子是复合函数时，忘记对其使用链式法则。

• Mistake: Losing a minus sign when differentiating trigonometric or exponential functions.
• 错误：对三角函数或指数函数求导时丢失负号。

To avoid these errors, always write u, v, u', v' explicitly before applying the rule. Check each derivative carefully.

为避免这些错误，应用法则前务必明确写出 u、v、u'、v'，并仔细检查每个导数。

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10. When to Use the Product Rule? | 何时使用乘积法则？

Use the product rule when you see a function that is the product of two (or more) different functions that cannot be easily combined into a single simpler expression.

当你看到两个（或多个）不同函数相乘，且无法轻松合并成一个更简单表达式时，就应使用乘积法则。

If the expression can be expanded easily, such as (x+1)(x+2), expanding first is often faster. However, for products like x³ ln x, the product rule is the only practical method.

如果表达式可以轻松展开，如 (x+1)(x+2)，先展开往往更快。然而对于 x³ ln x 这类乘积，乘积法则几乎是唯一实用方法。

Expression	Method	Reason
(x2+1)(x-3)	Expand first	Simple polynomials
x ex	Product rule	Different types of functions
x2 sin x	Product rule	Cannot expand

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11. Practice Questions | 练习问题

Try using the product rule to differentiate the following. Answers are shown at the end.

请尝试用乘积法则对下列函数求导。答案列在小节末尾。

• Question 1: y = (x² + 1)(2x - 3)
• 问题 1：y = (x² + 1)(2x - 3)

• Question 2: y = x³ ln x
• 问题 2：y = x³ ln x

• Question 3: y = sin x cos x
• 问题 3：y = sin x cos x

Answers:

答案：

1. 6x² - 6x + 2    2. x²(3ln x + 1)    3. cos 2x

For question 3, remember that ddx(sin x cos x) = sin x(-sin x) + cos x(cos x) = cos² x - sin² x = cos 2x.

对于问题 3，记住 ddx(sin x cos x) = sin x(-sin x) + cos x(cos x) = cos² x - sin² x = cos 2x。

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12. Summary | 总结

The product rule is a fundamental differentiation tool in A-Level Mathematics. It states that ddx(uv) = udvdx + vdudx.

乘积法则是 A-Level 数学中基本微分工具。它指出 ddx(uv) = udvdx + vdudx。

Remember to identify all factors, differentiate each one correctly, and combine the terms carefully. With practice, the product rule becomes second nature.

记住识别所有因子，正确求每个因子的导数，并仔细合并各项。多加练习，乘积法则就会变得得心应手。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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