📚 Exercise 7C: The Chain Rule | 练习7C：链式法则
Differentiation is one of the most powerful tools in the IB Mathematics syllabus, and the chain rule is the essential technique for handling composite functions. Exercise 7C focuses specifically on building fluency with this rule, from the simplest nested polynomials to intricate combinations involving exponentials, logarithms, and trigonometric expressions. Without a solid grasp of the chain rule, later topics such as implicit differentiation, related rates, and integration by substitution become significantly harder. This revision guide unpacks every angle of the chain rule as it appears in Exercise 7C, offering clear explanations, worked examples, and exam-targeted strategies.
微分是IB数学大纲中最有力的工具之一，而链式法则是处理复合函数的核心技巧。练习7C专门针对该法则的熟练运用，从最简单的嵌套多项式到涉及指数、对数和三角函数的复杂组合。如果没有扎实掌握链式法则，后续的隐函数求导、相关变化率以及换元积分等内容都将变得非常困难。这本复习指南从练习7C的每一个角度剖析链式法则，提供清晰的解释、典型示例和针对考试的解题策略。
1. Understanding the Chain Rule | 理解链式法则
The chain rule is used to differentiate a function of a function, that is, a composite function where one function is applied inside another. In simple terms, if you have an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This 'inside-out' approach is the heart of every question in Exercise 7C.
链式法则用于对复合函数求导，即一个函数嵌套在另一个函数内部的情况。简单来说，如果你有一个外部函数和一个内部函数，导数就等于外部函数在内部函数处的导数，乘以内部函数的导数。这种“由外而内”的求导思路是练习7C每一道题的核心。
). The outer function f receives the output of g, so the rate of change of y with respect to x depends first on how f responds to changes in g, and then on how g changes with x. This layered dependency is exactly what the chain rule captures.
为了直观理解，考虑 y = f(g(x))。外部函数 f 接收 g 的输出，因此 y 关于 x 的变化率首先取决于 f 对 g 变化的响应，然后再乘以 g 关于 x 的变化。这种层层递进的依赖正是链式法则所描述的。
2. The Formula and Notation | 公式与符号
The standard Leibniz notation for the chain rule is particularly helpful in Exercise 7C problems:
链式法则的标准莱布尼茨记法对练习7C的问题非常有帮助：
dy/dx = dy/du × du/dx
a simpler function of u. Many students prefer Lagrange notation: if h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). Both notations appear in IB mark schemes, so mastering both forms is essential.
这里 u 是所选择的内部函数，使得 y 成为 u 的更简单函数。许多学生偏爱拉格朗日记法：若 h(x) = f(g(x))，则 h'(x) = f'(g(x)) · g'(x)。这两种记法都出现在IB的评分方案中，因此掌握两种形式非常重要。
In Exercise 7C you will often need to explicitly identify u, compute du/dx, then find dy/du, and finally multiply the two. Writing out these steps systematically avoids sign errors and missing factors.
在练习7C中，你通常需要明确识别 u，计算 du/dx，然后求出 dy/du，最后两者相乘。系统地写出这些步骤可以避免符号错误和遗漏因子。
3. Differentiating Composite Functions | 复合函数的求导
⁴. Using the chain rule, dy/dx = 4u³ × (6x²) = 4(2x³ + 5)³ × 6x², which simplifies neatly.
复合函数本质上是外部函数包裹着内部函数。例如 y = (2x³ + 5)⁴ 的内部函数为 u = 2x³ + 5，外部函数为 y = u⁴。应用链式法则，dy/dx = 4u³ × (6x²) = 4(2x³ + 5)³ × 6x²，可以进一步整理。
Exercise 7C gradually increases complexity. First you deal with powers of linear functions, then quadratics, and eventually rational and radical functions. The key habit is always to ask: 'What is happening inside the brackets or under the radical?' That inner function determines du/dx.
练习7C的难度逐渐增加。首先处理线性函数的幂，然后是二次函数，最终涉及有理函数和根式函数。关键习惯是始终问自己：“括号内或根号下正在发生什么？”那个内部函数决定了 du/dx。
4. Applying the Chain Rule to Polynomial Powers | 链式法则在多项式幂中的应用
the form m(axⁿ + b)ᵐ⁻¹ × (naxⁿ⁻¹). It is easy to forget the factor from the inner function's derivative, especially when the inner function is more than a simple linear term.
练习7C中最常见的应用之一就是求 (axⁿ + b)ᵐ 这类表达式的导数。导数形式始终为 m(axⁿ + b)ᵐ⁻¹ × (naxⁿ⁻¹)。人们很容易忘记来自内部函数导数的因子，尤其当内部函数不仅仅是简单的线性项时。
2x - 3). Leaving the answer in this factorised form is often preferred in IB exams, as it explicitly shows the structure of the derivative.
考虑 y = (x² - 3x)⁵。令 u = x² - 3x，则 du/dx = 2x - 3。导数为 5u⁴ × du/dx，得到 dy/dx = 5(x² - 3x)⁴ × (2x - 3)。在IB考试中，通常更倾向于保留这种因式分解的形式，因为它明确展示了导数的结构。
5. Chain Rule with Trigonometric Functions | 链式法则与三角函数
erivative of the outer trig function must be multiplied by the derivative of the angle. Exercise 7C presents these alongside constants and coefficients to test your attention to detail.
三角函数非常频繁地被当作外部函数来处理。对于 y = sin(kx) 或 y = cos(ax + b)，外部三角函数的导数必须乘以角度的导数。练习7C将这些与常数和系数一起呈现，以考验你对细节的关注。
arly, y = cos(x²) gives dy/dx = -sin(x²) × 2x. Watch out for the negative sign from cosine's derivative—missing it is a classic error in Exercise 7C.
例如 y = tan(4x + 1)：外部导数为 sec²(4x + 1)，内部导数为 4，因此 dy/dx = 4 sec²(4x + 1)。类似地，y = cos(x²) 得到 dy/dx = -sin(x²) × 2x。注意余弦求导带来的负号——遗漏这一负号是练习7C中的一个经典错误。
6. Exponential and Logarithmic Functions | 指数与对数函数
Differentiating exponentials of the form eᶠ⁽ˣ⁾ requires the chain rule: the derivative remains eᶠ⁽ˣ⁾ multiplied by f'(x). For base-a exponentials aᶠ⁽ˣ⁾, the derivative is aᶠ⁽ˣ⁾ ln(a) × f'(x). Exercise 7C includes both types to ensure you can handle any base.
对 eᶠ⁽ˣ⁾ 形式的指数函数求导需要链式法则：导数仍为 eᶠ⁽ˣ⁾ 乘以 f'(x)。对于底数为 a 的指数函数 aᶠ⁽ˣ⁾，导数为 aᶠ⁽ˣ⁾ ln(a) × f'(x)。练习7C包含这两种类型，确保你能处理任意底数。
With logarithms, the derivative of ln(g(x)) is g' inner is g(x). For example, differentiating y = ln(5x² + 3) gives dy/dx = (10x)/(5x² + 3).
对于对数函数，ln(g(x)) 的导数为 g'(x)/g(x)。这一简洁的规则是链式法则的直接结果，其中外部函数为 ln，内部为 g(x)。例如，对 y = ln(5x² + 3) 求导得到 dy/dx = (10x)/(5x² + 3)。
7. Combining Rules: Product and Chain Rule | 法则组合：乘积法则与链式法则
Exercise 7C frequently challenges students with functions that require both the product rule and the chain rule. When a function is presented as a product of two expressions, and one or both factors themselves are composite, you must apply the chain rule within the product rule framework.
练习7C经常用需要同时使用乘积法则和链式法则的函数来挑战学生。当一个函数以两个表达式的乘积形式呈现，并且其中一个或两个因子本身就是复合函数时，你必须在乘积法则的框架内应用链式法则。
Take y = x³ · e²ˣ. The product rule gives dy/dx = derivative of x³ times e²ˣ plus x³ times derivative of e²ˣ. The derivative of e²ˣ uses the chain rule, giving 2e²ˣ. Thus dy/dx = 3x² · e²ˣ + x³ · 2e²ˣ. Factorising e²ˣ yiels e²ˣ x² (3 + 2x). Systematic working prevents messy mistakes.
以 y = x³ · e²ˣ 为例。乘积法则给出 dy/dx = x³ 的导数乘以 e²ˣ，加上 x³ 乘以 e²ˣ 的导数。e²ˣ 的导数使用链式法则，得到 2e²ˣ。因此 dy/dx = 3x² · e²ˣ + x³ · 2e²ˣ。提取 e²ˣ 得到 e²ˣ x² (3 + 2x)。系统性的解题步骤可以防止混乱的错误。
8. Higher-Order Derivatives | 高阶导数
Some Exercise 7C problems ask for the second derivative or ask you to show that a given function satisfies a differential equation. The chain rule must be applied repeatedly, and careful tracking of terms is required. The second derivative often involves product rule combinations of the first derivative's components.
练习7C中的某些问题会要求求二阶导数，或要求你证明给定函数满足某个微分方程。链式法则需要被反复应用，并且需要仔细追踪各项。二阶导数通常涉及对一阶导数各部分的乘积法则组合。
in(x²), then finding the second derivative requires the product rule: derivative of 2x times sin(x²) plus 2x times derivative of sin(x²), where differentiating sin(x²) again uses the chain rule. It is these layered applications that Exercise 7C aims to solidify.
如果一阶导数为 dy/dx = 2x sin(x²)，那么求二阶导数就需要乘积法则：2x 的导数乘以 sin(x²)，加上 2x 乘以 sin(x²) 的导数，而对 sin(x²) 求导又需使用链式法则。正是这种层层递进的应用，练习7C旨在巩固。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
The most frequent error in Exercise 7C is forgetting to multiply by the derivative of the inner function altogether. Students often differentiate the outer function correctly but stop there. Always ask: 'What is inside?' and explicitly find du/dx.
练习7C中最常见的错误是彻底忘记乘以内部函数的导数。学生经常正确地求出外部函数的导数，却止步于此。一定要问自己：“内部是什么？”并明确求出 du/dx。
x). Dropping the negative or misplacing the 2x is easy when rushing. Use brackets and check each factor.
另一个典型陷阱是负号和分式处理不当。对于 y = 1/(x² + 1)²，可以重写为 (x² + 1)⁻²，这样链式法则给出 -2(x² + 1)⁻³ × (2x)。匆忙中很容易遗漏负号或错放 2x。使用括号并检查每个因子可以有效避免。
- Always write u explicitly.
- Differentiate outer and inner separately before multiplying.
- Keep derivative results factorised unless instructed otherwise.
- Double-check trigonometric coefficients and signs.
- 始终明确写出 u。
- 在相乘之前分别对外部和内部求导。
- 除非另有要求，保持导数结果的因式分解形式。
- 仔细检查三角函数的系数和符号。
10. IB Exam-Style Questions | IB考试风格题目
ercise 7C mirrors the structured questions found in IB papers. A typical problem might be: 'Given f(x) = ln(cos(2x)), find f'(x) and hence evaluate f'(π/8).' You need the chain rule twice: derivative of cos(2x) is -2 sin(2x), so f'(x) = tan(2x) × (-2)? = -2 tan(2x). This kind of nested chain rule is a favourite in Section A and B questions.
练习7C模拟了IB试卷中的结构化问题。一个典型的题目可能是：“给定 f(x) = ln(cos(2x))，求 f'(x)，并据此计算 f'(π/8)。”你需要两次运用链式法则：cos(2x) 的导数是 -2 sin(2x)，因此 f'(x) = (1/cos(2x)) × (-sin(2x)) × 2 = -2 tan(2x)。这种嵌套的链式法则在试卷的A部分和B部分中备受青睐。
Another common style is the graphical interpretation: 'The graph of g(x) = e-x² has a maximum point. Find its coordinates.' You differentiate using the chain rule, set g' Exercise 7C trains the algebraic manipulation needed for these multi-step reasoning tasks.
另一种常见风格是图像解释：“函数 g(x) = e-x² 的图像有一个极大值点。求其坐标。”你需要使用链式法则求导，令 g'(x) = 0，然后求解。练习7C训练了这类多步骤推理任务所需的代数操作能力。
11. Tips for Mastering Exercise 7C | 掌握练习7C的技巧
Begin by categorising the functions: identify whether the main operation is a power, exponential, log, or trig. This determines your outer function immediately. Then isolate the inner function and write its derivative before combining. Chanting 'differentiate the outside, keep the inside, multiply by the derivative of the inside' can be a useful mantra.
从对函数进行分类开始：判断主要运算是幂函数、指数函数、对数函数还是三角函数。这会立刻确定你的外部函数。然后分离出内部函数，并在合并之前写出它的导数。默念“外函数求导，保持内函数不变，再乘以内函数的导数”是一个很有用的口诀。
Practice with varied examples that mix negative powers, fractional powers, and compound inner functions. The more automatic the chain rule becomes, the more mental space you free up for setting up bigger problems like optimisation and kinematics. Use coloured pens or highlighters to mark u and du/dx when reviewing mistakes.
用包含负指数、分数指数和复合内部函数的各种示例进行练习。链式法则越熟练，你在解决像最优化和运动学这类更大问题时就能释放出越多的思维空间。复习错题时可以使用彩色笔或荧光笔标记 u 和 du/dx。
| Inner Function (u) | Outer Function (y) | Derivative (dy/dx) |
|---|---|---|
| ax + b | uⁿ | n(ax+b)ⁿ⁻¹ × a |
| f(x) | eᵘ | eᶠ⁽ˣ⁾ × f'(x) |
| g(x) | ln(u) | g'(x)/g(x) |
| kx | sin(u) | k cos(kx) |
Use the table above as a quick reference when tackling the mixed exercises in the second half of Exercise 7C. Recognising these patterns transofrms the chain rule from a mechanical step into a strategic tool.
在处理练习7C后半部分的混合练习时，可将上表作为快速参考。识别这些模式能将链式法则从一个机械步骤转变为策略性工具。
12. Summary and Key Takeaways | 总结与要点
Exercise 7C builds the reflexive skill that underpins nearly all of calculus. The chain rule is not just a standalone technique; it integrates with product, quotient, and implicit differentiation, and it sets the stage for integration by substitution. Mastery here makes subsequent chapters significantly more manageable.
练习7C培养了支撑几乎所有微积分内容的下意识技能。链式法则不仅是一项独立的技术，它还与乘积法则、商法则和隐函数求导融为一体，并为换元积分法奠定基础。掌握这一部分内容会使得后续章节变得极大地易于应对。
By completing Exercise 7C thoroughly, you ensure that you can fluently differentiate composite functions, spot layered structures quickly, and avoid the small algebraic slips that cost marks. Keep practicing, trust the process, and the chain rule will soon become your sharpest differentiation tool.
通过全面完成练习7C，你能够确保流利地对复合函数求导，迅速识别嵌套结构，并避免那些导致丢分的小代数失误。坚持练习，相信这个过程，链式法则很快就会成为你最锋利的求导工具。
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