📚 Example 6.9.2: Differentiating y = x² sin x | 示例 6.9.2：对 y = x² sin x 求导
. This type of question appears regularly in Paper 1 Pure Mathematics under the topic of differentiation, and it tests your ability to select the correct rule, apply it accurately, and simplify the result.
在这个 AQA A-Level 数学例题中，我们使用乘积法则对 y = x² sin x 求导。此类题目经常出现在纯数学 1 卷的微分部分，考查学生选择正确法则、准确应用并化简结果的能力。
1. The Problem Statement | 题目陈述
/dx. In an AQA exam, this would typically be worth 4 marks, with the mark scheme rewarding the correct selection of the product rule, the accurate differentiation of each factor, and the final simplified expression.
已知函数 y = x² sin x，求 dy/dx。在 AQA 考试中，此题通常为 4 分，评分标准会奖励正确选择乘积法则、准确对每个因式求导以及最终化简后的表达式。
2. Required Knowledge: The Product Rule | 必备知识：乘积法则
u dv/dx. The rule is essential whenever two variable expressions are multiplied together.
乘积法则指出，如果 y = u v，其中 u 和 v 都是 x 的函数，那么导数为 dy/dx = u′ v + u v′。等价地，可以写作 dy/dx = v du/dx + u dv/dx。当两个含有变量的表达式相乘时，必须使用该法则。
A common misconception is to differentiate each factor separately and multiply the results. That is not correct. The product rule adds the two cross-terms together, which keeps the derivative accurate for all values of x.
一个常见的误区是分别对每个因式求导，然后将结果相乘。这是不正确的。乘积法则将两个交叉项相加，从而保持导数对所有 x 值都准确。
3. Step 1: Identify u and v | 第一步：确定 u 和 v
ice is convenient because both u and v have standard derivatives that are easy to compute.
我们将多项式因式设为 u，将三角函数因式设为 v。令 u = x²，v = sin x。这种选择很方便，因为 u 和 v 都有易于计算的标准导数。
² would still work with the product rule, because the rule is symmetric in form: u′ v + u v′ gives the same final result. However, choosing the polynomial as u keeps the differentiation simple.
虽然你也可以将 u 和 v 反过来设，即令 u = sin x、v = x²，乘积法则仍然适用，因为该法则形式对称：u′ v + u v′ 得到的最终结果相同。但将多项式设为 u 可以使求导过程更简单。
4. Step 2: Differentiate u and v | 第二步：对 u 和 v 求导
= 2x. This follows from the power rule: d/dx (xⁿ) = n xⁿ⁻¹. For n = 2, we obtain 2x¹ = 2x.
对 u = x² 求导，得到 du/dx = 2x。这由幂法则得出：d/dx (xⁿ) = n xⁿ⁻¹。当 n = 2 时，得到 2x¹ = 2x。
os x. This is one of the standard trigonometric derivatives that must be memorised for AQA A-Level Mathematics. The derivative of sin x is always cos x, never -cos x.
对 v = sin x 求导，得到 dv/dx = cos x。这是 AQA A-Level 数学必须记忆的标准三角函数导数之一。sin x 的导数始终是 cos x，而不是 -cos x。
5. Step 3: Apply the Product Rule | 第三步：应用乘积法则
he product rule formula dy/dx = u′ v + u v′. This gives dy/dx = (2x)(sin x) + (x²)(cos x).
将 u、v、du/dx 和 dv/dx 代入乘积法则公式 dy/dx = u′ v + u v′。得到 dy/dx = (2x)(sin x) + (x²)(cos x)。
2x sin x + x² cos x. This is the final answer for the given function, and both terms must be present for full marks.
因此，导数为 dy/dx = 2x sin x + x² cos x。这是给定函数的最终答案，两个项都必须出现才能获得满分。
6. Simplifying the Result | 化简结果
The expression 2x sin x + x² cos x is already in its simplest form as a sum of two terms. You could factor out a common factor x to write x(2 sin x + x cos x), but this factorised form is not usually required by AQA mark schemes unless the question explicitly asks for a fully factorised answer.
表达式 2x sin x + x² cos x 已经是两项之和的最简形式。你也可以提取公因式 x，写成 x(2 sin x + x cos x)，但除非题目明确要求完全因式分解，否则 AQA 评分标准通常不要求这种因式分解形式。
Leaving the answer as 2x sin x + x² cos x is perfectly acceptable and often preferred because it clearly shows the two terms produced by the product rule. Avoid over-simplifying in a way that introduces errors, such as attempting to combine terms that do not share a common factor.
将答案保留为 2x sin x + x² cos x 是完全可接受的，而且通常更受欢迎，因为它清楚地展示了乘积法则产生的两个项。避免过度化简而引入错误，例如试图合并不具有公因式的项。
7. Verification Using a Special Value | 使用特殊值进行验证
o zero in a smooth way, the tangent at x = 0 is horizontal, so the derivative should be 0.
一个有用的检查方法是，在原始函数梯度明显的点处计算导数值。在 x = 0 处，原函数 y = x² sin x 的值为 0，而且由于 x² 和 sin x 都以光滑的方式趋于零，x = 0 处的切线应该是水平的，因此导数应为 0。
os 0 = 0 + 0 = 0. This matches our expectation, so the result is consistent at this point.
将 x = 0 代入导数，得到 2(0) sin 0 + 0² cos 0 = 0 + 0 = 0。这与我们的预期一致，因此该结果在该点上是一致的。
8. Common Mistakes to Avoid | 常见错误
os x), which only accounts for the product of the two derivatives. This misses the term u′ v and is incorrect.
一个常见错误是忘记乘积法则，写成 dy/dx = (2x)(cos x)，这只是两个导数相乘。这漏掉了 u′ v 这一项，因此是错误的。
Another frequent mistake is differentiating sin x as -cos x. Remember that the derivative of sin x is +cos x, while the derivative of cos x is -sin x. Mixing up these signs can cost accuracy marks.
另一个常见错误是把 sin x 的导数写成 -cos x。请记住，sin x 的导数是 +cos x，而 cos x 的导数是 -sin x。混淆这些符号会失去答案分。
erivative is 2x, not x. Be careful to bring down the exponent and reduce the power by one.
第三个陷阱是错误应用幂法则。对于 u = x²，其导数是 2x，而不是 x。要小心地将指数移下来，并将幂降低一次。
9. AQA Mark Scheme Guidance | AQA 评分标准提示
In AQA mark schemes, you typically earn method marks for identifying u and v, for differentiating both parts correctly, and for substituting into the product rule formula. The final accuracy mark is awarded only when the derivative is fully correct and written in a clear form.
在 AQA 评分标准中，通常识别出 u 和 v、正确对两部分求导、并代入乘积法则公式即可获得方法分。只有当导数完全正确且书写清晰时，才能获得最终的答案分。
hod marks for the correct structure of the product rule, provided your working is clearly shown and the rest of the method is valid.
如果你犯了一个小错误，例如写成 dv/dx = -cos x，只要解题步骤清晰且方法其余部分有效，你仍然可能因乘积法则结构正确而获得方法分。
| Mark element | What earns the mark |
|---|---|
| M1 | Selects product rule and identifies u and v |
| M1 | Finds du/dx = 2x and dv/dx = cos x |
| M1 | Substitutes correctly into u′ v + u v′ |
| A1 | Obtains 2x sin x + x² cos x |
This table shows how the marks are typically allocated. Even if the final answer is wrong, clear method working can still earn most of the marks.
该表显示了分数的典型分配方式。即使最终答案错误，清晰的方法步骤仍然可以赢得大部分分数。
10. Practice Variations | 变式练习题
Try differentiating the following functions using the same product rule approach. These variations are similar to those found in AQA A-Level practice papers and exam-style questions.
尝试使用相同的乘积法则方法对以下函数求导。这些变式与 AQA A-Level 练习卷和考试题型中的题目类似。
y = x³ cos x | 对 y = x³ cos x 求导
y = ex sin x | 对 y = ex sin x 求导
y = ln x · sin x | 对 y = ln x · sin x 求导
y = x² tan x | 对 y = x² tan x 求导
v′. Check your answers by differentiating a second time or by substituting a simple value.
对于每一道题，识别 u 和 v，求出它们的导数，然后应用 dy/dx = u′ v + u v′。通过再次求导或代入一个简单数值来检查答案。
11. Summary of Key Points | 关键点总结
then dy/dx = u′ v + u v′. For y = x² sin x, the correct derivative is 2x sin x + x² cos x.
对乘积求导时，切勿直接对每个因式分别求导再相乘。务必使用乘积法则：如果 y = u v，则 dy/dx = u′ v + u v′。对于 y = x² sin x，正确的导数为 2x sin x + x² cos x。
Practise the standard derivatives of sin x and cos x until they are automatic, and always show your u, v, du/dx, and dv/dx before substituting into the rule. This not only helps you avoid mistakes but also maximises method marks in the exam.
反复练习 sin x 和 cos x 的标准导数直至熟练，并在代入法则前展示 u、v、du/dx 和 dv/dx。这不仅有助于避免错误，还能在考试中最大化方法分。
12. Conclusion | 结语
Mastering the product rule is essential for AQA A-Level Mathematics, as it appears frequently in differentiation questions involving trigonometric, exponential, and logarithmic functions. Example 6.9.2 provides a clear routine that you can apply to a wide range of problems.
掌握乘积法则对 AQA A-Level 数学至关重要，因为它在涉及三角函数、指数函数和对数函数的微分题中频繁出现。示例 6.9.2 提供了一个清晰的解题流程，你可以将其应用到各种问题中。
Work through the practice variations, revisit any mistakes, and remember that showing method is just as important as finding the final answer.
完成变式练习，回顾出现的错误，并记住展示解题过程与求出最终答案同样重要。
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