Exercise 9B.2: The Chain Rule | 练习9B.2：链式法则

📚 Exercise 9B.2: The Chain Rule | 练习9B.2：链式法则

Welcome to Exercise 9B.2, a focused practice set designed to build your confidence with one of the most essential tools in differential calculus: the chain rule. Whether you are preparing for your IB Analysis & Approaches or Applications & Interpretation course, mastering the chain rule is non-negotiable. This article will guide you through the key concepts, worked examples, common pitfalls, and progressive exercises that mirror the style of IB questions.

欢迎来到练习 9B.2，这是一套专为建立你对微分学最重要工具之一——链式法则的信心而设计的集中练习。无论你正在准备 IB 分析与方法还是应用与解释课程，掌握链式法则都是不折不扣的必选项。本文将引导你梳理核心概念、典型例题、常见错误，以及贴近 IB 题目风格的渐进式练习。

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1. The Chain Rule Formula | 链式法则公式

× du/dx. In Leibniz notation this relationship is elegant and easy to remember because the 'du' terms appear to cancel.

链式法则让我们能够对复合函数进行求导，也就是一个函数嵌套在另一个函数中的形式。如果 y = f(u) 且 u = g(x)，那么 y 关于 x 的导数就是 dy/dx = dy/du × du/dx。在莱布尼茨记号中，这种关系既优雅又容易记忆，因为 “du” 项似乎相互抵消了。

dy/dx = f′(g(x)) × g′(x)

: if h(x) = f(g(x)), then h′(x) = f′(g(x)) · g′(x). Many students find it helpful to think 'vative of the outside, keep the inside the same, then multiply by derivative of the inside'.

用函数记号通常表达为：若 h(x) = f(g(x))，则 h′(x) = f′(g(x)) · g′(x)。许多学生发现这样想很有用：“先对外层求导，里面保持不变，再乘以内层的导数”。

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2. Recognising Inner and Outer Functions | 识别内层和外层函数

ing the inner function correctly is crucial because you will need its derivative in the final multiplication.

应用链式法则的第一步是将复合函数分解为外层函数和内层函数。例如，在 y = (3x² + 5)⁴ 中，外层函数是 u⁴，内层函数是 u = 3x² + 5。正确识别内层函数至关重要，因为最后相乘时需要用到它的导数。

You can test your identification by mentally substituting u for the inner expression. The function should become purely dependent on u. Practice with expressions involving brackets, roots, exponentials, and trigonometric functions. If you see a function raised to a power, the power is usually the outer layer. If you see e raised to a function, the exponential is the outer operation.

你可以通过在脑海中将内层表达式替换为 u 来测试你的识别。函数应该变为完全依赖于 u。多练习涉及括号、根号、指数和三角函数的表达式。如果你看到一个函数带有幂次，幂次通常就是外层；如果你看到 e 的某函数次方，指数运算就是外层操作。

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3. Basic Worked Examples | 基础例题

Let' tice how the original inside (2x + 1) remains unchanged while we multiply by its derivative.

我们来用链式法则对 y = (2x + 1)⁵ 求导。设 u = 2x + 1，于是 y = u⁵。那么 dy/du = 5u⁴，而 du/dx = 2。相乘得到 dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴。注意内层 (2x + 1) 保持不变，同时乘以其导数。

Another example: differentiate y = √5x² − 3. Rewrite as y = (5x² − 3)^1/2. Let u = 5x² − 3, then y = u^1/2. dy/du = (1/2)u^−1/2 and du/dx = 10x. So dy/dx = (1/2)(5x² − 3)^−1/2 × 10x = 5x / √5x² − 3.

另一个例子：对 y = √5x² − 3 求导。将其改写为 y = (5x² − 3)^1/2。设 u = 5x² − 3，则 y = u^1/2。dy/du = (1/2)u^−1/2，du/dx = 10x。因此 dy/dx = (1/2)(5x² − 3)^−1/2 × 10x = 5x / √5x² − 3。

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4. Chain Rule with Trigonometric Functions | 三角函数与链式法则

3 = 3 cos(3x). Always check that the angle measure is in radians unless stated otherwise.

在 IB 练习中，三角函数经常以复合函数的形式出现。对于 y = sin(3x)，外层函数是 sin(u)，u = 3x。sin(u) 的导数是 cos(u)，而 du/dx = 3，因此 dy/dx = cos(3x) × 3 = 3 cos(3x)。除非另有说明，始终检查角度量是否以弧度为单位。

fies to −sin 2x. This shows how the chain rule often works together with trigonometric identities.

对于 y = cos²(x)，可以将其视为 y = (cos x)²。外层函数是 u²，内层是 u = cos x。求导：2u × (−sin x) = −2 cos x sin x，化简后得 −sin 2x。这体现了链式法则如何经常与三角恒等式协同工作。

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5. Exponential and Logarithmic Composite Functions | 指数与对数复合函数

e derivative of e to the power of a function is a classic chain rule application. If y = e^f(x), then dy/dx = e^f(x) ivative is itself, making the process especially tidy. For instance, differentiating y = e^4x², we get dy/dx = e^4x² × 8x = 8x e^4x².

e 的函数次方的导数是链式法则的经典应用。若 y = e^f(x)，则 dy/dx = e^f(x) × f′(x)。指数函数的独特之处在于其导数等于它本身，这使得过程特别简洁。例如，对 y = e^4x² 求导，得到 dy/dx = e^4x² × 8x = 8x e^4x²。

dy/dx = (3x² + 2) / (x³ + 2x). Memorising these patterns saves time and reduces mistakes.

对于自然对数函数，y = ln(g(x)) 给出 dy/dx = g′(x) / g(x)，这同样是链式法则的应用，ln(u) 的导数为 1/u 再乘以 du/dx。因此 y = ln(x³ + 2x) 得到 dy/dx = (3x² + 2) / (x³ + 2x)。记忆这些模式可以节省时间并减少错误。

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6. Combining the Chain Rule with Product and Quotient Rules | 链式法则结合乘积法则与商法则

v′ requires the chain rule: cos(5x) × 5 = 5 cos(5x). So dy/dx = 2x sin(5x) + 5x² cos(5x).

IB 题目经常要求你将链式法则与乘积法则或商法则结合使用。对于 y = x² sin(5x)，先用乘积法则：dy/dx = (第一个的导数)(第二个) + (第一个)(第二个的导数)。设 u = x²，v = sin(5x)。那么 u′ = 2x，而 v′ 需要链式法则：cos(5x) × 5 = 5 cos(5x)。因此 dy/dx = 2x sin(5x) + 5x² cos(5x)。

When applying the quotient rule, such as y = e^3x / (x² + 1), the chain rule is used when differentiating e^3x. The derivative is 3 e^3x. Then follow the quotient rule formula: (vu′ − uv′) / v². Organising your working systematically prevents sign errors and algebraic slips.

在应用商法则时，例如 y = e^3x / (x² + 1)，对 e^3x 求导就需要链式法则，其导数为 3 e^3x。然后按照商法则公式 (v u′ − u v′) / v² 继续。有条理地整理计算过程可以防止符号错误和代数失误。

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7. Repeated Chain Rule (Double Composition) | 多层链式法则（二重复合）

Some functions have more than one layer. To differentiate y = sin(e^x²), we write y = sin(u), u = e^v, v = x². Then dy/dx = dy/du × du/dv × dv/dx. This gives cos(u) × e^v × 2x = cos(e^x²) × e^x² × 2x. The chain rule extends naturally to any number of nested functions.

有些函数不止有一层。要对 y = sin(e^x²) 求导，我们可写作 y = sin(u)，u = e^v，v = x²。那么 dy/dx = dy/du × du/dv × dv/dx，得到 cos(u) × e^v × 2x = cos(e^x²) × e^x² × 2x。链式法则可以自然地扩展到任意多层的嵌套函数。

Keep the logic of working from the outside in. Each step you differentiate one layer, leaving the inside untouched, then multiply by the derivative of the inside. With practice, you can do this in a single line without intermediate substitutions, but beginners benefit from writing each substitution explicitly.

记住从外到内的逻辑。每一步你对外层求导，内层保持不变，然后乘以内层的导数。通过练习，你可以无需中间代换而一步完成，但初学者最好还是把每一步代换都明确写出来。

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

One of the most frequent errors is forgetting to multiply by the derivative of the inner function. Students differentiate the outer part correctly but stop short of the final multiplication. Always double-check: did you multiply by the derivative of what is inside the bracket or function?

最常见的错误之一就是忘记乘以内层函数的导数。学生正确地对外层求导，却停在了最后一步乘法之前。务必复查：你是否乘上了括号内或函数内部的导数？

t 1. Carelessness with signs costs many marks in IB assessments. Write out the inner function and its derivative separately if needed.

另一个错误是误判内层函数，特别是涉及负号或分式时。对于 y = (1 − x)³，内层函数是 u = 1 − x，其导数是 −1，而不是 1。在 IB 考试中，因粗心而出错的代价很高。如有需要，可单独写出内层函数及其导数。

Also, do not confuse the chain rule with the product rule. In (x² sin x), there is no composition; it is a product. Yet (sin x)² is a composition. Analyse the structure before you begin.

此外，不要将链式法则与乘积法则混淆。在 (x² sin x) 中没有复合关系，它是一个乘积。然而 (sin x)² 却是一个复合。在开始之前先分析结构。

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9. Applications to Tangents and Normals | 应用于切线与法线

gent at a point often requires the chain rule to compute the gradient. If the curve is y = √x² + 9, differentiate to find dy/dx = x / √x² + 9. At x = 4, the gradient is 4 / √25 = 0.8. Then use y − y₁ = m(x − x₁) to write the tangent equation.

求某点处的切线方程通常需要链式法则来计算斜率。若曲线为 y = √x² + 9，求导得 dy/dx = x / √x² + 9。在 x = 4 处，斜率为 4/√25 = 0.8。然后利用 y − y₁ = m(x − x₁) 写出切线方程。

Similarly, normal lines whose slope is the negative reciprocal of the tangent gradient can be found. These problems test both your differentiation and coordinate geometry skills, making them very common in IB Paper 1 and Paper 2.

类似地，法线斜率是切线斜率的负倒数。这类题目同时考查你的微分和坐标几何能力，因此在 IB 试卷一和试卷二中非常常见。

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10. Using the Chain Rule with Inverse Trigonometric Functions | 反三角函数与链式法则

an(u) = 1/(1 + u²). Applying the chain rule gives dy/dx = 1/(1 + (2x)²) × 2 = 2/(1 + 4x²).

像 arcsin、arccos 和 arctan 这样的反三角函数，当其自变量是 x 的函数时也需要链式法则。对 y = arctan(2x)，回忆 d/du arctan(u) = 1/(1 + u²)。应用链式法则得 dy/dx = 1/(1 + (2x)²) × 2 = 2/(1 + 4x²)。

Even though IB may not always heavily test inverse trig functions at Standard Level, the chain rule principle remains the same and prepares you for Higher Level topics such as integration by substitution backwards.

尽管 IB 标阶未必会重点考查反三角函数，但链式法则的原理不变，同时为高阶的专题（如反向换元积分法）做好准备。

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11. Chain Rule in Contextual Problems | 应用题中的链式法则

, dA/dr = 2π r, so dA/dt = 2π r × dr/dt. The chain rule bridges different rates of change.

相关变化率问题是链式法则在现实中的直接应用。若圆形油污的半径以恒定速率 dr/dt 增加，我们需要 dA/dt，则利用 dA/dt = dA/dr × dr/dt。因为 A = π r²，dA/dr = 2π r，因此 dA/dt = 2π r × dr/dt。链式法则连接了不同的变化率。

Other contextual problems include the growth of a population where the size depends on a function, or velocity and acceleration where position is given as a composite function of time. Recognising the chain rule structure in words is a key skill for IB's exploration and exam questions.

其他应用题包括人口增长中数量依赖于某个函数，或者位置为时间的复合函数时的速度和加速度。在文字题中识别链式法则的结构是 IB 探究和考试的关键技能。

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12. Summary and Practice Progression | 总结与练习进阶

Exercise 9B.2 systematically builds your chain rule skills from simple power composites to mixed rule problems. Start with easy polynomials inside brackets, then move to trigonometric, exponential, and logarithmic composites. Combine with product and quotient rules only when you are comfortable with the basic pattern.

练习 9B.2 系统地构建你的链式法则技能，从简单的幂函数复合开始，逐渐过渡到三角、指数和对数复合。只有在熟练基本模式之后，才去结合乘积法则和商法则进行练习。

Complete every problem in order, write out your u-substitutions clearly, and above all, always multiply by the derivative of the inside. With this revision guide you are well equipped to tackle any differentiation problem in IB Mathematics. Keep practising and gradually increase the complexity.

按顺序完成每一道题，清楚地写出你的 u 代换，最重要的是，永远记得乘以内层的导数。有了这份复习指南，你就有了应对 IB 数学中任何求导问题的充分准备。坚持练习，逐步增加难度。

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