📚 Example 5.2.3: Product Rule for y = x² sin x | 示例 5.2.3：对 y = x² sin x 使用乘积法则
ect to x. This is a standard AQA A-Level Mathematics question that tests your understanding of the product rule, a core technique in differentiation. We will break the process into clear steps so you can see exactly how to identify the two factors, apply the rule, simplify the result, and avoid common errors in an exam.
在本例题中，我们对函数 y = x² sin x 关于 x 求导。这是一道标准的 AQA A-Level 数学题，考查你对乘积法则这一核心微分技巧的理解。我们会把过程分解为清晰的步骤，让你清楚看到如何识别两个因子、应用法则、化简结果，并在考试中避免常见错误。
The product rule is used when a function is written as the product of two simpler functions. In this example, one factor is a polynomial and the other is a trigonometric function, so the rule is essential. After studying this example, you should be able to apply the same method to other products involving exponentials, logarithms, or trigonometric functions.
当一个函数可以写成两个较简单函数的乘积时，就需要使用乘积法则。在本例中，一个因子是多项式，另一个是三角函数，因此该法则必不可少。学完这个例子后，你应当能够将同样的方法应用于涉及指数函数、对数函数或三角函数的其他乘积。
1. The Product Rule Statement | 乘积法则陈述
If u(x) and v(x) are differentiable functions of x, then the derivative of their product is given by the formula below. This rule must be quoted accurately in AQA exams, as marks are often awarded for the correct statement before any substitution.
如果 u(x) 和 v(x) 都是关于 x 的可导函数，那么它们乘积的导数由下面的公式给出。在 AQA 考试中必须准确写出这条公式，因为在代入之前往往就会有公式表述分。
dy/dx = u dv/dx + v du/dx
Some textbooks write this as (fg)' = f 'g + fg' ame order throughout your working to reduce the chance of sign errors or copying mistakes.
有些教材将此写为 (fg)' = f 'g + fg'。两项的先后顺序不影响最终结果，但在整个计算过程中应保持一致的顺序，以减少符号错误或抄写错误的可能。
- English: u and v are the two factors in the original product.
- 中文：u 和 v 是原乘积中的两个因子。
- English: du/dx and dv/dx are the individual derivatives of u and v.
- 中文：du/dx 和 dv/dx 分别是 u 和 v 的导数。
- English: The product rule is not simply the product of the derivatives; the cross terms are essential.
- 中文：乘积法则并不是两个导数直接相乘；交叉项是必不可少的。
2. Step 1: Identify the Two Factors | 第 1 步：识别两个因子
atural choice is to let the polynomial part be u and the trigonometric part be v. This choice usually makes the later differentiation simpler because both factors have standard derivatives.
对于函数 y = x² sin x，最自然的选择是令多项式部分为 u，三角函数部分为 v。这种选择通常会使后续求导更简单，因为两个因子都有标准的导数公式。
u = x², v = sin x
ill give the same final answer. However, the standard convention is to choose u as the first factor and v as the second factor, which helps you organise your solution clearly.
如果你交换标签，令 u = sin x、v = x²，乘积法则仍然会得到相同的最终答案。不过，标准习惯是选择 u 作为第一个因子，v 作为第二个因子，这有助于清晰组织你的解题过程。
| Factor | Expression | Reason |
| u | x² | Polynomial factor, easy to differentiate |
| v | sin x | Trigonometric factor, standard derivative |
In an exam, you can write "Let u = x² and v = sin x" s your method clearly but also helps you recover marks if a later arithmetic error occurs.
在考试中，你可以一开始就写出 "Let u = x² and v = sin x"。这不仅能使你的方法清晰展示，而且如果后面出现算术错误，也有助于你获得方法分。
3. Step 2: Differentiate u and v | 第 2 步：对 u 和 v 分别求导
Now find the derivative of each factor separately. The derivative of x² is found using the power rule, and the derivative of sin x is a standard result that you should know by heart for AQA examinations.
现在分别求出每个因子的导数。x² 的导数使用幂法则求出，sin x 的导数是一个标准结果，你应该在 AQA 考试中熟记于心。
du/dx = 2x, dv/dx = cos x
- nglish: Differentiate u = x²: bring down the power 2 and reduce the power by 1, giving 2x.
- 中文：对 u = x² 求导：将指数 2 下移，指数减 1，得到 2x。
- English: Differentiate v = sin x: the derivative of sin x is cos x.
- 中文：对 v = sin x 求导：sin x 的导数是 cos x。
Keep these four pieces of information together: u, v, du/dx, and dv/dx. Many students lose marks because they differentiate correctly but then substitute the wrong expression into the formula, especially when the problem involves negative signs or fractional powers.
将这四个信息放在一起：u、v、du/dx 和 dv/dx。许多学生失分的原因是他们求导正确，但在代入公式时使用了错误的表达式，尤其是当题目涉及负号或分数指数时。
4. Step 3: Apply the Product Rule Formula | 第 3 步：应用乘积法则公式
Substitute u, v, du/dx, and dv/dx into the product rule formula. The first term is u times dv/dx, and the second term is v times du/dx. This gives the derivative before any simplification.
将 u、v、du/dx 和 dv/dx 代入乘积法则公式。第一项是 u 乘以 dv/dx，第二项是 v 乘以 du/dx。这样得到的是未化简的导数。
dy/dx = x²(cos x) + sin x(2x)
It is perfectly acceptable to leave the derivative in the form x² cos x + 2x sin x at this stage. In fact, writing the unsimplified expression first is good practice because it shows the examiner that you have applied the rule correctly.
在这一阶段，将导数写成 x² cos x + 2x sin x 的形式是完全可接受的。事实上，先写出未化简的表达式是良好的习惯，因为它向阅卷人表明你已经正确地应用了法则。
Always use brackets when substituting, especially for trigonometric functions such as cos x. Writing x² cos x is clear, but if the factor were something like x² ln x, brackets would help avoid confusion with the logarithm argument.
代入时一定要使用括号，尤其是对于 cos x 这类三角函数。写成 x² cos x 是清晰的，但如果因子是 x² ln x 这样的形式，括号就有助于避免与对数真数混淆。
5. Step 4: Simplify and Factor the Result | 第 4 步：化简并因式分解结果
The derivative obtained from the product rule can often be simplified by taking out a common factor. In this example, both terms contain an x, so we can factor x out to produce a cleaner final answer.
由乘积法则得到的导数通常可以通过提取公因式进行化简。在本例中，两项都含有 x，因此我们可以提取 x，得到一个更简洁的最终答案。
dy/dx = x(2 sin x + x cos x)
ctored form usually makes the arithmetic easier than using the expanded form, although both are mathematically equivalent.
有些题目可能还会要求你求某一点处的斜率，例如 x = π/2。代入因式分解后的形式通常比使用展开形式更容易计算，尽管两者在数学上是等价的。
- English: Expanded form: x² cos x + 2x sin x.
- 中文：展开形式：x² cos x + 2x sin x。
- English: Factored form: x(2 sin x + x cos x).
- 中文：因式分解形式：x(2 sin x + x cos x)。
In AQA mark schemes, either form is normally accepted unless the question specifically asks for a fully factorised answer. However, factorising can help you avoid errors when evaluating the derivative at a given x-value.
在 AQA 的评分标准中，两种形式通常都被接受，除非题目明确要求完全因式分解的答案。不过，因式分解有助于在给定 x 值处计算导数值时避免错误。
6. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One frequent mistake is to differentiate the product by simply multiplying the derivatives, that is, writing 2x cos x. This is incorrect because it ignores the two cross terms required by the product rule. Always remember that the product rule has two terms added together.
一个常见的错误是直接将两个导数相乘，即写成 2x cos x。这是错误的，因为它忽略了乘积法则所要求的两个交叉项。一定要记住，乘积法则是由两项相加组成的。
Another common error occurs when students mislabel u and v and then mix up the derivatives in the final substitution. To prevent this, write a small table or list before applying the formula, and tick off each element as you use it.
另一个常见错误发生在学生将 u 和 v 的标签标错，然后在最终代入时混淆了导数。为了防止这种情况，在应用公式之前先写一个小表格或列表，并在使用每个元素时打勾。
- English: Incorrect: dy/dx = (2x)(cos x) = 2x cos x.
- 中文：错误：dy/dx = (2x)(cos x) = 2x cos x。
- English: Correct: dy/dx = x² cos x + 2x sin x.
- 中文：正确：dy/dx = x² cos x + 2x sin x。
- English: Do not forget the v du/dx term; it is often the one students omit.
- 中文：不要忘记 v du/dx 这一项；它是学生经常遗漏的一项。
, then dv/dx would be -sin x, and the product rule would give x²(-sin x) + 2x cos x, which simplifies to -x² sin x + 2x cos x.
最后，在对三角函数求导时要小心符号。如果函数是 y = x² cos x，那么 dv/dx 就是 -sin x，乘积法则会给出 x²(-sin x) + 2x cos x，化简为 -x² sin x + 2x cos x。
7. Checking the Result by Differentiation from First Principles | 用第一性原理检验结果
n x, first principles would be lengthy, but the product rule provides a reliable shortcut.
虽然在 AQA 考试中通常不要求这样做，但你可以通过展开表达式或使用第一性原理来检验乘积法则的结果。对于 y = x² sin x，第一性原理会非常冗长，而乘积法则提供了一个可靠的捷径。
s 0) = 0. The original function y = x² sin x is flat at the origin, so this is consistent.
一种快速数值检验的方法是选择一个简单的值，例如 x = 0。在 x = 0 处，斜率应为 dy/dx = 0(2 sin 0 + 0 cos 0) = 0。原函数 y = x² sin x 在原点处是平坦的，因此这是吻合的。
You can also use technology such as a graphical calculator to plot the gradient function and compare it with your expression. This is useful for independent study, but in the actual exam you should rely on accurate symbolic differentiation rather than numerical checks.
你还可以使用图形计算器等工具绘制导函数图像，并与你的表达式进行比较。这对自主学习很有帮助，但在实际考试中，你应该依靠准确的符号求导，而不是数值检验。
8. Exam-Style Variations | 考试题型变式
AQA questions often extend this example by asking you to find the equation of the tangent or normal to the curve at a given point. To do this, you first find dy/dx using the product rule, then substitute the x-coordinate to obtain the gradient.
AQA 的题目经常在这个例子的基础上延伸，要求你求曲线在某一点处的切线或法线方程。为此，你首先使用乘积法则求出 dy/dx，然后代入 x 坐标得到斜率。
s the product rule with u = x² and v = sin(3x), where dv/dx = 3 cos(3x) by the chain rule.
另一种常见的变式是将乘积法则与链式法则结合使用。例如，对 y = x² sin(3x) 求导需要使用乘积法则，其中 u = x²、v = sin(3x)，而根据链式法则 dv/dx = 3 cos(3x)。
For y = x² sin(3x), dy/dx = x²(3 cos(3x)) + 2x sin(3x)
You may also see products involving exponential functions, such as y = x ex. In that case, letting u = x and v = ex gives du/dx = 1 and dv/dx = ex, so the derivative is ex + x ex, which can be factored as ex(1 + x).
你还可能遇到涉及指数函数的乘积，例如 y = x ex。此时令 u = x、v = ex，得到 du/dx = 1、dv/dx = ex，因此导数为 ex + x ex，可因式分解为 ex(1 + x)。
9. Practice Questions | 练习题
To consolidate your understanding, attempt the following questions. They follow the same pattern as Example 5.2.3 and will help you build confidence with the product rule in different contexts.
为了巩固你的理解，请尝试以下题目。它们遵循与示例 5.2.3 相同的模式，将帮助你建立在不同情境下使用乘积法则的信心。
- English: Differentiate y = x³ sin x with respect to x.
- 中文：对 y = x³ sin x 关于 x 求导。
- English: Differentiate y = x² cos x with respect to x.
- 中文：对 y = x² cos x 关于 x 求导。
- English: Differentiate y = x ex with respect to x.
- 中文：对 y = x ex 关于 x 求导。
- English: Find the gradient of y = x² sin x at x = π/4.
- 中文：求 y = x² sin x 在 x = π/4 处的斜率。
Work through each question by identifying u and v, finding their derivatives, applying the product rule, and simplifying. Check your answers carefully, especially when trigonometric values such as sin(π/4) and cos(π/4) appear.
完成每道题时，请识别 u 和 v，求出它们的导数，应用乘积法则，并进行化简。仔细检查答案，尤其是当出现 sin(π/4) 和 cos(π/4) 这类三角函数值时。
10. Summary and Key Takeaways | 总结与关键要点
ntiate y = x² sin x by setting u = x² and v = sin x, then using dy/dx = u dv/dx + v du/dx.
乘积法则是一项基本的微分技巧，在 AQA A-Level 数学中你必须能够快速准确地应用。示例 5.2.3 展示了如何通过令 u = x²、v = sin x，然后使用 dy/dx = u dv/dx + v du/dx 来对 y = x² sin x 求导。
dy/dx = 2x sin x + x² cos x = x(2 sin x + x cos x)
Remember to write the rule clearly, label your factors, differentiate each one separately, substitute carefully, and simplify when appropriate. Avoid multiplying the two derivatives directly, and always watch for negative signs in trigonometric derivatives.
请记住：清晰地写出法则，标记因子，分别求导，仔细代入，并在适当的时候化简。避免直接将两个导数相乘，并且始终注意三角函数导数中的负号。
With regular practice, the product rule will become second nature, and you will be able to handle exam questions that combine it with the chain rule, implicit differentiation, or applications to tangents and normals.
通过经常练习，乘积法则将成为你的第二天性，你将能够应对将它与链式法则、隐函数微分或切线与法线应用相结合的考试题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图