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。

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 \(f(x) = x^2 \sin x\), the function \(f\) is the product of \(x^2\) and \(\sin x\).

乘积就是两个或多个表达式相乘的结果。例如，在 \(f(x) = x^2 \sin x\) 中，函数 \(f\) 是 \(x^2\) 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。

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：

\(\frac{d}{dx}[u v] = u \frac{dv}{dx} + v \frac{du}{dx}\)

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 \(\frac{d}{dx}(uv) = u v’ + v u’\). Both forms are identical.

你也可能看到写作 \(\frac{d}{dx}(uv) = u v’ + v u’\)，这两种形式完全相同。

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 \(\Delta x\), leading to changes \(\Delta u\) and \(\Delta v\).

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 \(\Delta x\)，相应地引起 \(\Delta u\) 和 \(\Delta v\)。

Then \(\Delta(uv) = (u+\Delta u)(v+\Delta v) – uv = u\Delta v + v\Delta u + \Delta u \Delta v\). Dividing by \(\Delta x\) and letting \(\Delta x \to 0\) gives the formula.

于是 \(\Delta(uv) = (u+\Delta u)(v+\Delta v) – uv = u\Delta v + v\Delta u + \Delta u \Delta v\)。除以 \(\Delta x\) 并令 \(\Delta x \to 0\)，就得到公式。

The term \(\Delta u \Delta v\) disappears because it is negligible compared with the linear terms as \(\Delta x \to 0\).

其中 \(\Delta u \Delta v\) 这一项在 \(\Delta x \to 0\) 时与线性项相比可以忽略不计。

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

Differentiate \(y = (3x^2 + 1)(2x – 5)\).

求 \(y = (3x^2 + 1)(2x – 5)\) 的导数。

Let \(u = 3x^2 + 1\) and \(v = 2x – 5\). Then \(\frac{du}{dx} = 6x\) and \(\frac{dv}{dx} = 2\).

令 \(u = 3x^2 + 1\)，\(v = 2x – 5\)。则 \(\frac{du}{dx} = 6x\)，\(\frac{dv}{dx} = 2\)。

Using the product rule:

应用乘积法则：

\(\frac{dy}{dx} = (3x^2 + 1)(2) + (2x – 5)(6x)\)

Simplify: \(\frac{dy}{dx} = 6x^2 + 2 + 12x^2 – 30x = 18x^2 – 30x + 2\).

化简：\(\frac{dy}{dx} = 6x^2 + 2 + 12x^2 – 30x = 18x^2 – 30x + 2\)。

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。

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

Differentiate \(y = x^2 \sin x\).

求 \(y = x^2 \sin x\) 的导数。

Let \(u = x^2\) and \(v = \sin x\). Then \(\frac{du}{dx} = 2x\) and \(\frac{dv}{dx} = \cos x\).

令 \(u = x^2\)，\(v = \sin x\)。则 \(\frac{du}{dx} = 2x\)，\(\frac{dv}{dx} = \cos x\)。

Apply the product rule:

应用乘积法则：

\(\frac{dy}{dx} = x^2 \cos x + 2x \sin x\)

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

可以写成 \(x(x\cos x + 2\sin x)\)。

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。

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 \(\frac{du}{dx} = 1\), and to find \(\frac{dv}{dx}\) we use the chain rule: \(\frac{d}{dx} e^{3x} = 3e^{3x}\).

令 \(u = x\)，\(v = e^{3x}\)。则 \(\frac{du}{dx} = 1\)，而求 \(\frac{dv}{dx}\) 需要用到链式法则：\(\frac{d}{dx} e^{3x} = 3e^{3x}\)。

So:

因此：

\(\frac{dy}{dx} = x \cdot 3e^{3x} + e^{3x} \cdot 1 = e^{3x}(3x + 1)\)

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。

7. Products with Logarithms | 含对数的乘积

Consider \(y = \ln x \cdot \cos x\). Here \(u = \ln x\) and \(v = \cos x\).

考虑 \(y = \ln x \cdot \cos x\)。这里 \(u = \ln x\)，\(v = \cos x\)。

We have \(\frac{du}{dx} = \frac{1}{x}\) and \(\frac{dv}{dx} = -\sin x\).

我们有 \(\frac{du}{dx} = \frac{1}{x}\)，\(\frac{dv}{dx} = -\sin x\)。

Thus:

因此：

\(\frac{dy}{dx} = \ln x (-\sin x) + \cos x \left(\frac{1}{x}\right) = \frac{\cos x}{x} – \sin x \ln x\)

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。

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 \(y = u v w\)，

\(\frac{dy}{dx} = u v w’ + u w v’ + v w u’\)

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, \(y = x^2 \sin x \cos x\). Differentiate each factor:

例如，\(y = x^2 \sin x \cos x\)。对每个因子分别求导：

\(\frac{dy}{dx} = x^2 \sin x (-\sin x) + x^2 \cos x (\cos x) + 2x \sin x \cos x\)

Then simplify using trigonometric identities if required.

然后根据需要利用三角恒等式进行化简。

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

Mistake: Writing \(\frac{dy}{dx} = u’ v’\) instead of \(u v’ + v u’\).
错误：写成 \(\frac{dy}{dx} = u’ v’\)，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。

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 \(x^3 \ln x\), 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 \(x^3 \ln x\) 这类乘积，乘积法则几乎是唯一实用方法。

Expression	Method	Reason
\((x^2+1)(x-3)\)	Expand first	Simple polynomials
\(x e^x\)	Product rule	Different types of functions
\(x^2 \sin x\)	Product rule	Cannot expand

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: \(y = (x^2 + 1)(2x – 3)\)
问题 1：\(y = (x^2 + 1)(2x – 3)\)

Question 2: \(y = x^3 \ln x\)
问题 2：\(y = x^3 \ln x\)

Question 3: \(y = \sin x \cos x\)
问题 3：\(y = \sin x \cos x\)

Answers:

答案：

1. \(6x^2 – 6x + 2\) 2. \(x^2(3\ln x + 1)\) 3. \(\cos 2x\)

For question 3, remember that \(\frac{d}{dx}(\sin x \cos x) = \sin x(-\sin x) + \cos x(\cos x) = \cos^2 x – \sin^2 x = \cos 2x\).

对于问题 3，记住 \(\frac{d}{dx}(\sin x \cos x) = \sin x(-\sin x) + \cos x(\cos x) = \cos^2 x – \sin^2 x = \cos 2x\)。

12. Summary | 总结

The product rule is a fundamental differentiation tool in A-Level Mathematics. It states that \(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}\).

乘积法则是 A-Level 数学中基本微分工具。它指出 \(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}\)。

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。

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The Product Rule | 乘积法则（微分法）