📚 IB Math Exercise 14B.1: Mastering the Chain Rule | IB数学练习14B.1：精通链式法则
In the IB Mathematics: Analysis and Approaches syllabus, Exercise 14B.1 is a fundamental set of problems designed to reinforce your understanding of the chain rule—one of the most essential differentiation techniques. This article will walk you through the key concepts, worked examples, and common mistakes associated with these exercises, ensuring you can tackle them with confidence.
在IB数学分析与方法的课程中，练习14B.1是一组旨在巩固你对链式法则理解的基础习题。链式法则是最重要的微分技巧之一。本文将引导你掌握与此练习相关的核心概念、范例和常见错误，帮助你自信地应对这些问题。
1. Understanding the Chain Rule | 理解链式法则
e derivative of the inner function. Symbolically: if y = f(u) and u = g(x), then dy/dx = f'(u) x g'(x).
链式法则用于对复合函数求导，即形式为 y = f(g(x)) 的函数。它指出，导数 dy/dx 等于外层函数的导数（保持内层函数不变）乘以内层函数的导数。用符号表示：若 y = f(u) 且 u = g(x)，则 dy/dx = f'(u) x g'(x)。
Chain Rule: dy/dx = f'(g(x)) · g'(x)
2. The Structure of Exercise 14B.1 | 练习14B.1的结构
Exercise 14B.1 in the standard IB textbook typically presents a sequence of differentiation problems that gradually increase in complexity. You will start with simple power functions like (3x + 1)⁵ and move on to trigonometric, exponential, and logarithmic composites. The goal is to automate the 'differentiate the outside, keep the inside, then multiply by the derivative of the inside' thought process.
标准IB教材中的练习14B.1通常提供一系列难度逐步递增的求导问题。你会从 (3x + 1)⁵ 这样的简单幂函数开始，然后过渡到三角函数、指数函数和对数的复合函数。其目标是让你自动化“外导乘内导”的思维过程：先对外层求导并保持内层不变，再乘以内层的导数。
3. Basic Power Chain: Differentiate (ax + b)ⁿ | 基本幂链式：求导 (ax + b)ⁿ
ve of u⁴ is 4u³, and the derivative of u is 5. Thus, dy/dx = 4(5x - 2)³ · 5 = 20(5x - 2)³.
考虑例子：y = (5x - 2)⁴。识别外层函数为 u⁴，内层函数为 u = 5x - 2。u⁴ 的导数是 4u³，u 的导数是 5。因此，dy/dx = 4(5x - 2)³ · 5 = 20(5x - 2)³。
If y = (ax + b)ⁿ, then y' = a n (ax + b)ⁿ⁻¹
- Example from 14B.1: Differentiate y = (2x + 7)⁶ → y' = 12(2x + 7)⁵
- 练习14B.1习题：求导 y = (2x + 7)⁶ → y' = 12(2x + 7)⁵
4. Trigonometric Chain Rule | 三角链式法则
s k·cos(kx). For cos(kx), it becomes -k·sin(kx). Exercise 14B.1 reinforces these patterns.
当内层函数为线性函数时，对三角函数求导十分直接。对于 y = sin(kx)，导数为 k·cos(kx)。对于 cos(kx)，则变为 -k·sin(kx)。练习14B.1强化了这些模式。
d/dx [sin(ax + b)] = a·cos(ax + b)
- y = cos(3x) → y' = -3 sin(3x)
- y = sin(4x - 1) → y' = 4 cos(4x - 1)
- y = tan(2x) → sec²(2x) · 2 = 2 sec²(2x) (using derivative of tan u = sec²u · u')
Make sure you handle the angle in radians, as IB examinations assume radian measure.
请确保角度以弧度制处理，因为IB考试默认使用弧度制。
5. Exponential Chain Rule | 指数链式法则
The derivative of eᵘ is eᵘ · u'. Since eˣ ltiplies by the derivative of the exponent. For base a (a > 0), d/dx [aᵘ] = aᵘ · ln(a) · u'.
eᵘ 的导数是 eᵘ · u'。因为 eˣ 的导数等于它本身，链式法则只需乘以指数的导数即可。对于底数 a (a > 0)，d/dx [aᵘ] = aᵘ · ln(a) · u'。
d/dx [e3x+2] = 3e3x+2
- y = e5x → y' = 5e5x
- y = 24x → y' = 4·24x·ln 2
- y = esin x → y' = esin x · cos x (composite within a composite)
In Exercise 14B.1 you will practice these with simple linear exponents first, then more embedded functions.
在练习14B.1中，你将首先练习线性指数的这些情况，然后涉及更深的嵌套函数。
6. Logarithmic Chain Rule | 对数链式法则
For natural logs, d/dx [ln(u)] = u' oblems often have positive arguments. For logarithms with general base: d/dx [logₐ(u)] = u' / (u ln a).
对于自然对数，d/dx [ln(u)] = u' / u。如果需要绝对值请记住，但IB题目通常保证真数为正。对于一般底数的对数：d/dx [logₐ(u)] = u' / (u ln a)。
d/dx [ln(2x+1)] = 2 / (2x+1)
- y = ln(5x - 3) → y' = 5 / (5x - 3)
- y = log₁₀(x²) → y' = (2x) / (x² ln 10)
7. Chain Rule with Multiple Compositions | 多重复合的链式法则
nner structure and require repeated application of the chain rule. For instance, y = sin(e2x). Here, outer is sin, middle is eu, inner is 2x. The derivative becomes cos(e2x) · e2x · 2.
练习14B.1中的某些题目会超越简单的外内结构，需要重复应用链式法则。例如，y = sin(e2x)。这里，外层是 sin，中间层是 eu，最内层是 2x。导数变为 cos(e2x) · e2x · 2。
y = √(tan(x)) → y' = (1/(2√(tan(x)))) · sec²(x)
Write it as (tan x)1/2: derivative = 1/2(tan x)-1/2 · sec² x. Always break down the composition stepwise.
将其写作 (tan x)1/2：导数为 1/2(tan x)-1/2 · sec² x。务必逐步拆解复合层。
8. Common Pitfalls in Exercise 14B.1 | 练习14B.1的常见误区
Mistake 1: Forgetting the derivative of the inner function. ome students write 3(3x²)², missing the inner derivative 6x. Correct: 3(3x²)² · 6x = 54x⁵.
错误1：忘记乘以内层函数的导数。 对于 y = (3x²)³，有些学生写成 3(3x²)²，遗漏了
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