📚 Example 5.5.3: Chain Rule with Composite Functions | 例题 5.5.3：复合函数的链式法则
le is designed to consolidate your understanding of how to identify the inner and outer functions before applying the chain rule formula.
链式法则是 AQA A-Level 数学中最常考查的技能之一。例题 5.5.3 重点求解形如 y = (3x² + 5x - 1)⁴ 的复合函数的导数。该例题旨在巩固你在应用链式法则公式之前，如何识别内层函数与外层函数。
1. Context and Learning Objectives | 背景与学习目标
This example appears in the differentiation chapter after the basic rules for powers, exponentials and trigonometric functions have been introduced. By the end of this worked example, you should be able to decompose a composite function, differentiate the outer and inner parts separately, and combine the results to obtain dy/dx.
本例题出现在微分章节中，位于幂函数、指数函数和三角函数的基本求导法则之后。通过这个例题，你应当能够分解复合函数，分别对外层和内层部分求导，并将结果组合得到 dy/dx。
The specific learning outcomes covered here are directly aligned with the AQA specification for differentiation of composite functions. In the exam, this skill is often tested as a short independent question or as part of a larger problem involving tangents, stationary points or rates of change.
此处涵盖的具体学习目标与 AQA 大纲中复合函数求导的要求直接对应。在考试中，这一技能通常作为独立的短题考查，或作为涉及切线、驻点或变化率的更大问题的一部分出现。
- Identify the structure u = g(x) inside y = f(u)
- Apply dy/dx = dy/du × du/dx correctly
- Simplify the final derivative without expanding unnecessarily
- Recognise when the chain rule must be combined with the product or quotient rule
识别 y = f(u) 中的结构 u = g(x)；正确应用 dy/dx = dy/du × du/dx；简化最终导数而不必进行不必要的展开；识别何时需要将链式法则与乘法法则或除法法则结合使用。
2. The Core Rule: Chain Rule Recap | 核心法则：链式法则回顾
ith respect to x is given by the product of two derivatives. The idea is that y changes with u, u changes with x, so the total rate of change is the product of the two individual rates.
对于任意可微函数 y = f(u) 和 u = g(x)，y 关于 x 的导数由两个导数的乘积给出。其思想是 y 随 u 变化，u 随 x 变化，因此总变化率是两个单独变化率的乘积。
dy/dx = dy/du × du/dx
An equivalent form uses the notation f'(g(x)) × g' use it keeps the two differentiation steps clearly separated until the final substitution.
另一种等价形式使用记号 f'(g(x)) × g'(x)。在实践中，许多学生发现使用中间变量 u 更容易，因为它能将两个求导步骤清晰分开，直到最后代换。
The chain rule is not just a formula to memorise; it is the mechanism that makes differentiation of complicated expressions possible. Once you master this pattern, you can differentiate polynomials raised to powers, trigonometric functions with linear or non-linear arguments, exponentials, logarithms, and combinations of these.
链式法则不仅仅是一个需要记忆的公式；它是使复杂表达式求导成为可能的机制。一旦掌握这一模式，你就能对高次多项式、具有线性或非线性内部变量的三角函数、指数函数、对数函数以及它们的组合进行求导。
3. Worked Example 5.5.3 Step 1: Identify Outer and Inner Functions | 例题 5.5.3 第一步：识别外层与内层函数
1. The outer function is the power of 4, and the inner function is the quadratic expression inside the brackets.
给定 y = (3x² + 5x - 1)⁴，我们首先将其明确改写为 y = u⁴，其中 u = 3x² + 5x - 1。外层函数是四次幂，内层函数是括号内的二次表达式。
ood check is to ask: if I plug u into the outer function, do I get the original expression? Here, replacing u with 3x² + 5x - 1 in y = u⁴ gives exactly the original y.
选择正确的内层函数是最重要的一步。如果选错了 u，后面的解答都会出错。一个好的检验方法是问自己：如果我把 u 代入外层函数，是否能得到原表达式？这里，在 y = u⁴ 中用 3x² + 5x - 1 替换 u，恰好得到原来的 y。
u = 3x² + 5x - 1, y = u⁴
4. Step 2: Differentiate Outer and Inner Functions | 第二步：分别求外层与内层函数的导数
dx = 6x + 5. It is essential to keep the variables of differentiation clear at this stage.
对外层函数关于 u 求导。由于 y = u⁴，幂法则给出 dy/du = 4u³。然后对内层函数关于 x 求导：u = 3x² + 5x - 1，因此 du/dx = 6x + 5。现阶段保持求导变量清晰至关重要。
A common error is to differentiate the outer function with respect to x instead of u, but that is not correct because the chain rule separates the two roles. The outer derivative must always be taken with respect to the inner variable u, and the inner derivative with respect to the original variable x.
一个常见错误是对外层函数关于 x 而不是关于 u 求导，但这是不正确的，因为链式法则将两个角色分开。外层导数必须始终对内部变量 u 求导，而内层导数对原始变量 x 求导。
dy/du = 4u³, du/dx = 6x + 5
5. Step 3: Apply the Chain Rule Formula | 第三步：应用链式法则公式
). We then replace u with the original inner function to express the answer in terms of x.
将两个导数代入链式法则公式 dy/dx = dy/du × du/dx。得到 dy/dx = 4u³ × (6x + 5)。然后将 u 替换为原来的内层函数，用 x 表示答案。
Performing the substitution gives the final derivative in factored form. This is the answer that should be written on the answer line unless the question specifically asks for an expanded form.
进行代换后得到因式分解形式的最终导数。除非题目特别要求展开形式，否则应在答案线上写出这个结果。
dy/dx = 4(3x² + 5x - 1)³ (6x + 5)
At this point, you have completed the core differentiation. The remaining work is only simplification or rearrangement, depending on the question.
至此，你已经完成了核心求导。剩下的工作只是根据题目要求进行化简或整理。
6. Simplifying the Derivative | 化简导数表达式
The final expression can be left in factored form; AQA mark schemes accept this. Factored form is usually preferred because it shows the structure clearly and saves time. It is also much easier to check for errors when the derivative retains the original bracket and the inner derivative as a visible factor.
最终表达式可以保持因式分解形式；AQA 评分标准接受这种写法。因式分解形式通常更受欢迎，因为它清晰地展示了结构并节省时间。当导数保留原括号和内层导数的可见因子时，检查错误也容易得多。
If you choose to expand, you would multiply 4(3x² + 5x - 1)³ by (6x + 5), which produces a lengthy polynomial of degree 7. This is rarely necessary and increases the chance of arithmetic mistakes. Unless the question explicitly says "give your answer in its simplest expanded form", leave it factored.
如果你选择展开，就需要将 4(3x² + 5x - 1)³ 乘以 (6x + 5)，得到一个冗长的七次多项式。这很少有必要，而且增加了算术错误的机会。除非题目明确说 "以最简展开形式给出答案"，否则保持因式分解。
)³ × 4 is equivalent, but the conventional form places the constant and the bracket first.
导数也可以将常数因子 4 与括号放在一起，或者交换因子的顺序。例如 (6x + 5)(3x² + 5x - 1)³ × 4 是等价的，但习惯上把常数和括号放在前面。
7. Common Mistakes to Avoid | 常见错误与规避
Many students forget to multiply by du/dx, leaving the derivative as 4(3x² + 5x - 1)³. This is the single most common error in chain rule questions. Others differentiate the outer power incorrectly or misidentify the inner function.
许多学生忘记乘以 du/dx，将导数写成 4(3x² + 5x - 1)³。这是链式法则题目中最常见的单一错误。另一些学生对外层幂求导错误，或错误识别内层函数。
Always check that the final derivative contains the derivative of the inner function as a factor. If your answer does not include (6x + 5) somewhere, you have likely missed the second part of the chain rule. Also verify that the power has decreased from 4 to 3 on the bracket.
务必检查最终导数中是否包含内层函数的导数作为因子。如果你的答案中没有出现 (6x + 5)，很可能漏掉了链式法则的第二部分。还要确认括号上的幂已从 4 降到 3。
- Forgetting the du/dx factor
- Misreading the inner function
- Confusing dy/du with du/dx
- Applying the power rule twice to the inner function
- Incorrectly subtracting 1 from the outer power
忘记 du/dx 因子；错误识别内层函数；混淆 dy/du 与 du/dx；对内层函数重复使用幂法则；外层幂减 1 错误。
8. Extension: Chain Rule with Trigonometric and Exponential Functions | 拓展：三角函数与指数函数的链式法则
The same method applies to y = sin(2x + 1), y = e3x + 1, then dy/du = cos(u) and du/dx = 2, giving dy/dx = 2cos(2x + 1). Similarly, for y = e3x, we get dy/dx = 3e3x.
同样的方法适用于 y = sin(2x + 1)、y = e3x 和 y = ln(5x - 2)。例如，若 y = sin(u) 且 u = 2x + 1，则 dy/du = cos(u) 且 du/dx = 2，从而 dy/dx = 2cos(2x + 1)。类似地，对于 y = e3x，可得 dy/dx = 3e3x。
These standard results can be summarised in a short table. The key pattern is that the derivative of the inner function becomes a multiplying factor in front of the trigonometric, exponential or logarithmic expression.
这些标准结果可以用一个简短的表格总结。关键模式是内层函数的导数成为三角函数、指数函数或对数
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