📚 Exercise 16C: Differentiation of Trigonometric Functions and the Chain Rule | 练习16C：三角函数微分与链式法则
Exercise 16C in the IB Mathematics: Analysis and Approaches course is designed to deepen your understanding of differentiation, specifically targeting trigonometric functions, the chain rule, and often their extension into implicit differentiation and related rates. These exercises move beyond simple power‑rule derivatives and require you to recognise when a function is composed of an outer and an inner function, or when an equation defines a relationship implicitly. Mastering these skills is not only essential for achieving high marks in Paper 1 and Paper 2 but also builds the foundation for topics like optimisation, kinematics, and differential equations later in the syllabus.
IB数学分析与方法课程中的练习16C旨在加深你对微分的理解，尤其侧重于三角函数、链式法则，并常延伸到隐函数微分和相关变化率。这些练习超越了简单的幂法则求导，要求你识别出函数何时由外层和内层复合而成，或者方程何时以隐式定义关系。掌握这些技能不仅对在试卷一和试卷二中取得高分至关重要，而且为后续大纲中的优化、运动学和微分方程等内容奠定了坚实的基础。
1. Introduction to Exercise 16C | 练习16C简介
Typical Exercise 16C problems in IB textbooks involve differentiating expressions like sin(3x), cos²(x), or tan(√x 5 that require implicit differentiation, and later you will apply these techniques to real‑ related rates. This article walks you through the core concepts, common pitfalls, and exam‑ready strategies that will turn Exercise 16C from a challenge into a confident routine.
IB教材中典型的练习16C题目涉及对诸如 sin(3x)、cos²(x) 或 tan(√x) 等表达式进行微分，你需要将标准三角函数的导数知识与链式法则结合起来。随着学习的深入，你会遇到像 x² + y² = 25 这样需要隐函数微分的方程，之后还会通过相关变化率将这些技巧应用于现实情境。本文将带你梳理核心概念、常见错误和备考策略，把练习16C从挑战变成你从容应对的常规操作。
2. Review of Basic Differentiation Rules | 基本微分规则复习
Power Rule: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.
幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。
Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
Constant Multiple Rule: d/dx [c · f(x)] = c · f'(x).
常数倍法则：d/dx [c · f(x)] = c · f'(x)。
Derivative of eˣ: d/dx (eˣ) = eˣ.
eˣ 的导数：d/dx (eˣ) = eˣ。
Derivative of ln x: d/dx (ln x) = 1/x, for x > 0.
ln x 的导数：d/dx (ln x) = 1/x，其中 x > 0。
Before tackling Exercise 16C, ensure you can differentiate any power function, exponential, and logarithmic expression instantly. These rules will be called upon repeatedly when applying the chain rule.
在攻克练习16C之前，请确保你能熟练地对任何幂函数、指数函数和对数函数求导。在应用链式法则时，这些规则会被反复调用。
3. The Chain Rule Explained | 链式法则详解
chain rule is the backbone of Exercise 16C. It states that if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx).
链式法则是练习16C的基石。它指出，若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。用莱布尼茨记号表达，若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) · (du/dx)。
A key insight is to identify the 'inner function' u⁵ and the inner function is u = 3x² + 1. Hence dy/dx = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴.
关键在于识别出‘内层函数’u 并单独对其求导。对于 y = (3x² + 1)⁵，外层函数是 u⁵，内层函数为 u = 3x² + 1。因此 dy/dx = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴。
Memorise this pattern: differentiate the outer function, leave the inner unchanged, then multiply by the derivative of the inner function. This 'outside‑inside' mantra will prevent many sign errors.
牢记这个模式：对外层函数求导，保持内层不变，再乘以内层函数的导数。这个‘先外后内’的口诀可以避免许多符号错误。
4. Differentiating Trigonometric Functions | 三角函数的微分
The derivatives of the six basic trigonometric functions must be memorised exactly. A handy table summarises them:
六个基本三角函数的导数必须准确记忆。下面这个方便的表格进行了总结：
| Function f(x) | Derivative f'(x) |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| cot x | -csc² x |
| sec x | sec x tan x |
| csc x | -csc x cot x |
中文对照：sin x 的导数是 cos x；cos x 的导数是 -sin x；tan x 的导数是 sec² x；cot x 的导数是 -csc² x；sec x 的导数是 sec x tan x；csc x 的导数是 -csc x cot x。
Notice the sneaky negative sign that appears in the derivatives of cos, cot, and csc functions. A large portion of Exercise 16C mistakes come from omitting that negative sign, especially when the chain rule wraps a negative inside a composite function.
注意 cos、cot 和 csc 函数的导数中悄悄出现的负号。练习16C中很大一部分错误正是由于遗漏了这个负号，尤其是当链式法则将负号包裹在复合函数内部时。
5. Chain Rule with Trigonometric Functions | 链式法则与三角函数结合
the chain rule becomes essential. For g(x) = sin(ax + b), the derivative is a cos(ax + b).
当三角函数的自变量不单单是 x，而是像 3x 这样的线性函数或更复杂的表达式时，链式法则就不可或缺。对于 g(x) = sin(ax + b)，其导数为 a cos(ax + b)。
n y = tan u. dy/du = sec² u and du/dx = 10x. Thus dy/dx = sec²(5x²) · 10x = 10x sec²(5x²).
例题：对 y = tan(5x²) 求导。设 u = 5x²，则 y = tan u。dy/du = sec² u，du/dx = 10x。于是 dy/dx = sec²(5x²) · 10x = 10x sec²(5x²)。
cos² x sin x. Always differentiate the outer power first, preserve the trigonometric part, and then multiply by the derivative of the trigonometric inner core.
当遇到三角函数的幂，例如 y = cos³(x) 时，将其重写为 y = (cos x)³ 并应用链式法则：dy/dx = 3(cos x)² · (-sin x) = -3 cos² x sin x。始终首先对外层幂次求导，保留三角函数部分，然后乘以三角函数内芯的导数。
6. Implicit Differentiation | 隐函数微分
te every term with respect to x, treating y as a function of x and using the chain rule on y terms.
隐函数微分用来处理 y 未显式表达为 x 的函数的方程，例如圆 x² + y² = r²。要求出 dy/dx，需对每一项关于 x 求导，把 y 当作 x 的函数，并对含 y 的项使用链式法则。
ht side r² differentiates to 0. Hence 2x + 2y (dy/dx) = 0, and solving gives dy/dx = -x/y.
x² 求导得 2x。y² 求导得 2y (dy/dx)。右边 r² 求导为 0。所以 2x + 2y (dy/dx) = 0，解出 dy/dx = -x/y。
Exercise 16C may also include curves like eʸ + sin y = x³. Differentiate: eʸ isolate it. Never forget to multiply by dy/dx whenever you differentiate a function of y.
练习16C可能还包括曲线 eʸ + sin y = x³。求导：eʸ (dy/dx) + cos y (dy/dx) = 3x²，提取 dy/dx 并解出它。每当对 y 的函数求导时，千万不要忘记乘上 dy/dx。
7. The Derivative of Inverse Trigonometric Functions | 反三角函数的导数
In some IB syllabi, Exercise 16C extends to the derivatives of inverse trigonometric functions. The key results are:
在一些IB大纲中，练习16C会延伸到反三角函数的导数。核心结果如下：
d/dx (arcsin x) = 1 / √1 - x², for |x| < 1.
d/dx (arcsin x) = 1 / √1 - x²，当 |x| < 1。
d/dx (arccos x) = -1 / √1 - x².
d/dx (arccos x) = -1 / √1 - x²。
d/dx (arctan x) = 1 / (1 + x²).
d/dx (arctan x) = 1 / (1 + x²)。
/(1+4x²). The structure is identical to previous chain rule applications, but the arithmetic requires careful handling of denominators.
当内部自变量为函数时，如 arctan(2x)，需使用链式法则：导数 = (1/(1+(2x)²)) · 2 = 2/(1+4x²)。其结构与前面的链式法则应用完全相同，只是代数运算需要谨慎处理分母。
8. Related Rates – Connecting Derivatives to Real Life | 相关变化率——导数实际应用
Related rates problems are a natural extension of implicit differentiation and the chain rule. In these scenarios, two or more quantities vary with time, and you are given the rate of change of one quantity, then asked to find the rate of change of another.
相关变化率问题是隐函数微分和链式法则的自然延伸。在这些情景里，两个或更多的量随时间变化，已知其中一个量的变化率，要求找出另一个量的变化率。
om the wall at 0.5 m/s, how fast does the top descend when the bottom is 3 m from the wall? ter substituting the instantaneous distances and the known horizontal speed. The chain rule thus transforms a geometric static equation into a dynamic model.
一个经典例子：一把 5 米长的梯子斜靠在墙上。如果底端以 0.5 m/s 的速度滑离墙壁，当底端离墙 3 米时，顶端下降的速度是多少？这里的关系式 x² + y² = 25 要对时间 t 求导：2x (dx/dt) + 2y (dy/dt) = 0。代入瞬时距离和已知水平速度后，解出 dy/dt。链式法则因此将一个几何静态方程转变成了动态模型。
9. Common Mistakes in Exercise 16C | 练习16C常见错误
Forgetting the inner derivative: When differentiating sin(4x), writing cos(4x) without the factor 4 is the most frequent slip. Always check: did I multiply by the derivative of the angle?
遗漏内层导数：对 sin(4x) 求导时，写成 cos(4x) 而不带因子 4 是最常见的疏漏。始终检查：我是否乘上了角度的导数？
Trigonometric sign errors: Mixing up the derivative of cos x (which gives -sin x) with that of sin x easily happens under exam pressure. Recite the negative ones aloud when practicing.
三角函数符号错误：在考试压力下很容易把 cos x 的导数（得到 -sin x）与 sin x 的导数混淆。练习时大声念出带负号的那些。
Neglecting dy/dx in implicit differentiation: Differentiating y³ yields 3y² but the chain rule demands an extra dy/dx factor. Omission instantly invalidates your solution.
隐函数微分时忽略 dy/dx：对 y³ 求导得到 3y²，但链式法则要求额外乘上 dy/dx。遗漏此因子会立即导致解答无效。
Misapplying product rule in combination: When an expression involves both a product and a trigonometric function, e.g. x sin(2x), differentiate systematically with the product rule while applying the chain rule to sin(2x).
结合使用时误用乘积法则：当表达式同时涉及乘积和三角函数，例如 x sin(2x)，需系统性地使用乘积法则，同时对 sin(2x) 运用链式法则。
10. Worked Examples from Exercise 16C | 练习16C典型例题
Example 1: Find f'(x) if f(x) = cos(3x² + 1).
Solution: f'(x) = -sin(3x² + 1) · (6x) = -6x sin(3x² + 1).
例题1：若 f(x) = cos(3x² + 1)，求 f'(x)。
解答：f'(x) = -sin(3x² + 1) · (6x) = -6x sin(3x²
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