📚 Exercise 14B: The Chain Rule | 链式法则练习
In IB Mathematics, Exercise 14B often focuses on one of the foundational calculus techniques: the chain rule for differentiation. This exercise guides students through nested functions where the derivative is not immediately obvious, requiring a systematic approach to unlock correct answers. Mastering the chain rule is essential not only for this topic but for tackling rates of change, implicit differentiation, and related rates that appear later in the syllabus.
在 IB 数学中，练习 14B 通常聚焦于微积分的基本技巧之一：导数的链式法则。这个练习引导学生处理复合函数，这类函数的导数并不显而易见，需要系统的方法才能得出正确答案。掌握链式法则不仅对这一主题至关重要，也为应对后续教学大纲中的变化率、隐函数微分和相关变化率问题打下坚实基础。
1. What Is the Chain Rule? | 什么是链式法则？
If y = f(g(x)), the chain rule states that the derivative dy/dx = f'(g(x)) · g' ifferentiate the outer function first, keeping the inner function unchanged, and then multiply by the derivative of the inner function.
如果 y = f(g(x))，链式法则指出导数 dy/dx = f'(g(x)) · g'(x)。用莱布尼茨标记法可写作 dy/dx = dy/du · du/dx，其中 u = g(x) 是中间函数。链式法则的本质是先对外层函数求导，内层函数保持不变，然后再乘以内层函数的导数。
- Identify the outside function and the inside function.
- Alternate notation: If y = (u(x))ⁿ, then dy/dx = n · u(x)ⁿ⁻¹ · u'(x).
- 识别出外层函数和内层函数。
- 另一种表示法：若 y = (u(x))ⁿ，则 dy/dx = n · u(x)ⁿ⁻¹ · u'(x)。
2. Why 'Chain'? | 为什么叫“链式”？
The name 'chain rule' comes from the way derivatives link together like a chain. If y depends on u, and u depends on x, then y also depends on x through this chain of dependencies. The derivative dy/dx is the product of two rates of change: dy/du (sensitivity of y to u) and du/dx (sensitivity of u to x).
“链式法则”得名于导数像链条一样环环相扣。如果 y 依赖于 u，u 依赖于 x，那么 y 通过这一依赖链条也依赖于 x。导数 dy/dx 是两个变化率的乘积：dy/du（y 对 u 的敏感度）和 du/dx（u 对 x 的敏感度）。
In physics, this is analogous to multiplying two conversion factors. For example, if you know kilometers per hour and hours per minute, you can chain them to find kilometers per minute.
在物理学中，这类似于两个换算因子的相乘。例如，如果你知道每小时多少公里和每分钟多少小时，就可以通过链式相乘得出每分钟多少公里。
3. Simple Power-Chain Examples | 简单的幂函数链式法则示例
Consider y = (3x² + 5x)⁴. The outer function is 'something to the power 4' (6x + 5). You must write the derivative of the inner function separately and multiply it.
考虑 y = (3x² + 5x)⁴。外层函数是“某数的 4 次方”，内层函数是 3x² + 5x。应用链式法则：dy/dx = 4(3x² + 5x)³ · (6x + 5)。必须单独写出内层函数的导数并与之相乘。
A common mistake is forgetting to multiply by the derivative of the inside, leaving the answer as merely 4(3x² + 5x)³, which is incomplete.
一个常见错误是忘记乘以内层函数的导数，仅将答案写成 4(3x² + 5x)³，这样是不完整的。
4. Trigonometric Functions and the Chain Rule | 三角函数与链式法则
in(x²) · 2x = -2x sin(x²). Always leave the inner angle unchanged in the outer derivative.
当对线性或更复杂内层函数的三角函数求导时，链式法则不可或缺。对于 y = sin(2x + 1)，导数为 cos(2x + 1) · 2 = 2 cos(2x + 1)。对于 y = cos(x²)，可得 dy/dx = -sin(x²) · 2x = -2x sin(x²)。务必在外层导数中保持内层角度不变。
| y = tan(u) | dy/dx = sec²(u) · u' |
| y = csc(u) | dy/dx = -csc(u)cot(u) · u' |
上表显示了正切和余切函数的链式求导公式，务必牢记内层函数的导数因子。
5. Exponential and Logarithmic Chain Applications | 指数与对数函数的链式应用
nctions with base e, the chain rule is elegant: if y = e^(u(x)), then dy/dx = e^(u(x)) · u'(x). For a general base a, y = a^(u(x)) gives dy/dx = a^(u(x)) · ln a · u'(x).
对于以 e 为底的指数函数，链式法则简洁优美：若 y = e^(u(x))，则 dy/dx = e^(u(x)) · u'(x)。对于一般底数 a，y = a^(u(x)) 的导数为 dy/dx = a^(u(x)) · ln a · u'(x)。
differentiation also relies on the chain rule. If y = ln(u(x)), then dy/dx = (1/u(x)) · u'(x). Example: y = ln(5x³ − 2x) → dy/dx = (15x² − 2) / (5x³ − 2x).
对数微分同样依赖链式法则。若 y = ln(u(x))，则 dy/dx = (1/u(x)) · u'(x)。例如：y = ln(5x³ − 2x) → dy/dx = (15x² − 2) / (5x³ − 2x)。
6. Double and Triple Chain Applications | 双重与三重链式法则
ion is composed of three nested functions, the chain rule extends naturally. For y = sin³(√x), rewrite as y = [sin(x1/2)]³. Then dy/dx = 3[sin(x1/2)]² · cos(x1/2) · (1/2)x-1/2. You differentiate from the outermost layer inward, multiplying derivatives along the way.
当函数由三层嵌套组成时，链式法则自然推广。对于 y = sin³(√x)，可重写为 y = [sin(x1/2)]³。则 dy/dx = 3[sin(x1/2)]² · cos(x1/2) · (1/2)x-1/2。从最外层向内逐层求导，沿途将各层导数相乘。
The general pattern is: derivative = (outer' evaluated at middle) · (middle' evaluated at inner) · (inner'). No matter how many layers, this multiplicative chain holds.
通用模式为：导数 = (外层函数在中间层的导数) · (中间层在内层的导数) · (内层的导数)。无论有多少层，这一乘法链始终成立。
7. The Chain Rule with Inverse Trigonometric Functions | 反三角函数与链式法则
standard, but the chain rule adds an inner derivative. For y = arcsin(u(x)), dy/dx = (1 / √1 − u²) · u'. Similarly, y = arctan(u(x)) gives dy/dx = (1 / (1 + u²)) · u'.
反三角函数在 IB 练习中频繁出现。其导数公式标准，但链式法则会增加一个内层导数。对于 y = arcsin(u(x))，dy/dx = (1 / √1 − u²) · u'。类似地，y = arctan(u(x)) 的导数为 dy/dx = (1 / (1 + u²)) · u'。
Example: y = arcsin(3x). Then dy/dx = (1 / √1 − 9x²) · 3 = 3 / √1 − 9x². Always check the domain restrictions when working with inverse trig functions.
示例：y = arcsin(3x)。则 dy/dx = (1 / √1 − 9x²) · 3 = 3 / √1 − 9x²。处理反三角函数时，务必检查定义域限制。
8. Implicit Differentiation Using the Chain Rule | 用链式法则进行隐函数微分
2x + 2y · dy/dx = 0. The term 2y · dy/dx comes from the chain rule: d(y²)/dx = 2y · dy/dx.
隐函数微分将 y 视作 x 的函数，并在每次对含 y 的表达式求导时应用链式法则。若给出 x² + y² = 25，两边对 x 求导得 2x + 2y · dy/dx = 0。项 2y · dy/dx 正是来自链式法则：d(y²)/dx = 2y · dy/dx。
Without the chain rule, implicit differentiation would be impossible. This technique is heavily tested in Exercise 14B and beyond.
没有链式法则，隐函数微分便无从进行。这一技巧在练习 14B 及后续内容中是高频考点。
9. Common Mistakes and How to Avoid Them | 常见错误及应对方法
The most frequent error is neglecting to multiply by the inner derivative. Always ask: 'What is the derivative of the inside?' and write it down explicitly. Another pitfall is misidentifying the inner and outer functions, especially when square roots, fractions, and powers overlap.
最常见的错误是忘记乘以内层导数。要始终问自己：“内层的导数是什么？”并明确写下。另一个易错点是在平方根、分式和幂次叠加时错误识别内外层函数。
For y = √2x+1, rewrite as (2x+1)1/2. Outer is power 1/2, inner is 2x+1. Then dy/dx = (1/2)(2x+1)-1/2 · 2 = 1 / √2x+1.
对于 y = √2x+1，应重写为 (2x+1)1/2。外层是 1/2 次方，内层是 2x+1。然后 dy/dx = (1/2)(2x+1)-1/2 · 2 = 1 / √2x+1。
Also, be careful with signs. The derivative of cos(u) is −sin(u) · u'; the negative sign is often omitted.
此外，要留意符号。cos(u) 的导数是 −sin(u) · u'；负号常被遗漏。
10. Practice Technique and Examination Tips | 练习技巧与应试诀窍
When working through Exercise 14B, start by clearly labelling u (the inner function). Work step by step: (1) Identify u, (2) Find du/dx, (3) Express y in terms of u, (4) Find dy/du, (5) Multiply to get dy/dx. Use brackets liberally to avoid algebraic slip-ups.
做练习 14B 时，起步要清晰标出 u（内层函数）。按步骤推进：(1) 确定 u，(2) 求 du/dx，(3) 用 u 表示 y，(4) 求 dy/du，(5) 相乘得 dy/dx。多使用括号以避免代数疏漏。
In exams, even if you forget a formula, you can derive it using the chain rule logic. This robust method works for any differentiable composition. Always present your final answer in a simplified form, as IB mark schemes often reward simplification.
考试中，即使忘记了某个公式，你也可以借助链式法则的逻辑推导出来。这一稳健方法适用于任何可微的复合函数。始终以简化形式呈现最终答案，因为 IB 评分方案通常对化简有加分。
11. Connections to Other Topics | 与其他主题的关联
The chain rule is not isolated; it underpins related rates, differential equations, and integration by substitution. For instance, the reverse of the chain rule is used in u-substitution integration. Understanding the chain rule deeply will make integration by reversing it far more intuitive.
链式法则并非孤岛；它是相关变化率、微分方程以及换元积分法的基础。例如，链式法则的逆过程就用于 u-代换积分法。深入理解链式法则，将使逆用它的积分法直观得多。
Also, in probability density functions and maximum likelihood estimation, the chain rule appears when differentiating composite likelihood functions.
此外，在概率密度函数和极大似然估计中，对复合似然函数求导时也会出现链式法则。
12. Summary and Final Advice | 总结与最后建议
Mastering the chain rule is a non-negotiable milestone in IB Calculus. Exercise 14B offers extensive practice on this critical skill. Approach each problem with the outer‑inner decomposition, and gradually you will perform the steps mentally with speed and accuracy.
掌握链式法则是 IB 微积分中不可逾越的里程碑。练习 14B 为这一关键技能提供了大量练习。处理每一道题时都采用外-内层分解法，逐步你将能够在脑中快速、准确地完成每一步骤。
Remember: 'Differentiate outside, multiply by derivative of inside'—this simple mantra will guide you through the most daunting of composite functions.
牢记：“外导内不变，乘以内导数”——这一简单口诀将引领你攻克最纷繁的复合函数。
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