📚 The Chain Rule | 链式法则
The chain rule is a fundamental tool in calculus used to differentiate composite functions. It allows you to break down complicated expressions into simpler parts, finding the derivative of the outer function multiplied by the derivative of the inner function. In A-Level Edexcel Mathematics, mastering the chain rule is essential for tackling a wide range of problems, from basic polynomial compositions to implicit and parametric differentiation.
链式法则是微积分中用于对复合函数求导的基本工具。它可以将复杂的表达式拆分为更简单的部分，用外层函数的导数乘以内层函数的导数来求得结果。在 A-Level Edexcel 数学中，掌握链式法则对于解决从基本多项式复合到隐函数和参数方程求导的各类问题至关重要。
1. What is the Chain Rule? | 什么是链式法则？
erivative of y with respect to u multiplied by the derivative of u with respect to x. This can be written as dy/dx = dy/du × du/dx, or in function notation, if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).
如果 y 是 u 的函数，而 u 是 x 的函数，那么 y 就是 x 的复合函数。链式法则指出，y 对 x 的导数等于 y 对 u 的导数乘以 u 对 x 的导数。这可以写作 dy/dx = dy/du × du/dx，或者用函数符号表示：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。
2. Leibniz Notation for the Chain Rule | 链式法则的莱布尼茨记法
Leibniz notation is particularly intuitive for the chain rule because the differentials 'cancel' visual aid helps students remember to multiply the two rates of change, but it is important to treat it as a formal rule rather than a literal fraction cancellation.
莱布尼茨记法对链式法则来说非常直观，因为微分符号可以像分数一样“约分”。如果 y = y(u) 且 u = u(x)，那么 dy/dx = (dy/du) × (du/dx)。这种视觉化的辅助手段有助于学生记住要将两个变化率相乘，但必须将其视为正式的求导法则，而不是真正意义上的分数约分。
3. Chain Rule with Power Functions | 链式法则与幂函数
The most common application is differentiating expressions like (ax + b)n. By letting the inner function u = ax + b and the outer function y = un, the derivative is dy/dx = n un-1 × a. For example, if y = (2x + 1)5, then dy/dx = 5(2x + 1)4 × 2 = 10(2x + 1)4. Always remember to multiply by the derivative of the inside term.
最常见的应用是对 (ax + b)n 这类表达式求导。设内层函数 u = ax + b，外层函数 y = un，则导数为 dy/dx = n un-1 × a。例如，若 y = (2x + 1)5，则 dy/dx = 5(2x + 1)4 × 2 = 10(2x + 1)4。务必记住乘以内层表达式的导数。
d/dx ( (3x - 4)7 ) = 7(3x - 4)6 × 3 = 21(3x - 4)6
4. Chain Rule with Trigonometric Functions | 三角函数的链式法则
sin(kx) = k cos(kx), and d/dx cos(kx) = -k sin(kx). For composite trig functions like sin2(x) or (sin x)2, rewrite as (sin x)2 and apply the chain rule: derivative is 2 sin x × cos x = sin 2x.
在对线性表达式的三角函数求导时，将角度视为内层函数。例如，d/dx sin(kx) = k cos(kx)，d/dx cos(kx) = -k sin(kx)。对于形如 sin2(x) 或 (sin x)2 的复合三角函数，可改写为 (sin x)2 并应用链式法则：导数为 2 sin x × cos x = sin 2x。
d/dx tan(5x) = 5 sec²(5x)
5. Chain Rule with Exponential and Logarithmic Functions | 指数和对数函数的链式法则
The derivative of eu is eu u/dx. For natural logarithms, d/dx ln(u) = (1/u) × du/dx. This extends to any base: d/dx au = au ln a × du/dx. A typical Edexcel question might ask for the derivative of e2x2 + 1; applying the chain rule gives dy/dx = e2x2 + 1 × 4x.
eu 的导数为 eu × du/dx。对于自然对数，d/dx ln(u) = (1/u) × du/dx。这可以推广到任意底数：d/dx au = au ln a × du/dx。典型的 Edexcel 题目可能要求对 e2x2 + 1 求导；应用链式法则得到 dy/dx = e2x2 + 1 × 4x。
d/dx ln(3x + 5) = 3/(3x + 5)
6. Multiple Applications of the Chain Rule | 多次使用链式法则
metimes a function is a composition of three or more functions. For instance, y = ln(sin(x2)). Here, let u = x2, v = sin u, and y = ln v. Then dy/dx = (1/v) × cos u × 2x = (cos(x2)/sin(x2)) × 2x = 2x cot(x2). The chain rule extends naturally: multiply all the derivative 'links' together.
有时函数是三个或更多函数的复合。例如 y = ln(sin(x2))。设 u = x2，v = sin u，y = ln v。那么 dy/dx = (1/v) × cos u × 2x = (cos(x2)/sin(x2)) × 2x = 2x cot(x2)。链式法则可以自然扩展：将所有导数“环节”相乘即可。
7. Combining Chain Rule with Product Rule | 链式法则与乘积法则结合
You often need to combine the chain rule and the product rule when differentiating expressions like x2 (2x+1)3. First, identify the product: u = x2, v = (2x+1)3. Then u' = 2x, while v' requires the chain rule: v' = 3(2x+1)2 × 2 = 6(2x+1)2. Finally, apply the product rule: dy/dx = u'v + uv' = 2x(2x+1)3 + x2 × 6(2x+1)2, which can be simplified by factoring out common terms.
在求导如 x2 (2x+1)3 这类表达式时，常常需要结合链式法则和乘积法则。首先识别乘积：u = x2，v = (2x+1)3。则 u' = 2x，而 v' 需要用链式法则：v' = 3(2x+1)2 × 2 = 6(2x+1)2。最后应用乘积法则：dy/dx = u'v + uv' = 2x(2x+1)3 + x2 × 6(2x+1)2，通过提取公因式可进一步化简。
8. Chain Rule in Implicit Differentiation | 隐函数求导中的链式法则
Implicit differentiation relies heavily on the chain rule. When differentiating an equation like x2 + y2 = 25 with respect to x, treat y as a function of x. The derivative of y2 use of the chain rule. Solving for dy/dx yields -x/y. Every time you differentiate a term involving y, you must multiply by dy/dx.
隐函数求导非常依赖链式法则。当对方程 x2 + y2 = 25 关于 x 求导时，将 y 视为 x 的函数。y2 的导数为 2y × dy/dx。这个 dy/dx 的出现正是因为链式法则。解出 dy/dx 得 -x/y。每次对包含 y 的项求导时，都必须乘以 dy/dx。
9. Parametric Differentiation | 参数方程求导
dy/dx = dy/dt × dt/dx, and since dt/dx = 1/(dx/dt), the result follows. In exam questions, you often need to find the equation of a tangent or a normal at a specific parameter value.
对于参数方程 x = f(t) 和 y = g(t)，梯度 dy/dx 由 (dy/dt) / (dx/dt) 求得。这个公式直接来自链式法则：dy/dx = dy/dt × dt/dx，而 dt/dx = 1/(dx/dt)，因此得到该结果。在考试题中，常常需要求在特定参数值处的切线或法线方程。
10. Recognising the Reverse Chain Rule in Integration | 识别积分中的反向链式法则
topic, the chain rule also helps with integration. The reverse chain rule states that ∫ f'(g(x)) g'(x) dx = f(g(x)) + C. Recognising integrands of the form k g'(x) [g(x)]n or g'(x)/g(x) allows for quick integration. For example, ∫ 2x (x2 + 1)3 dx is exactly the derivative of (1/4)(x2+1)4, which can be found by reversing the chain rule.
虽然链式法则主要是求导的主题，它也对积分有帮助。反向链式法则指出 ∫ f'(g(x)) g'(x) dx = f(g(x)) + C。识别出形如 k g'(x) [g(x)]n 或 g'(x)/g(x) 的被积函数可以实现快速积分。例如，∫ 2x (x2 + 1)3 dx 正好是 (1/4)(x2+1)4 的导数，这可以通过倒推链式法则得到。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Students often forget to multiply by the derivative of the inner function, especially when the inner function is more than a simple linear term. Another common error is misapplying the chain rule when using the product or quotient rule together. Always identify the 'layers' of composition explicitly: write out u substitution if needed. Double-check that you have multiplied by du/dx before simplifying.
学生经常忘记乘以内层函数的导数，尤其是当内层函数不仅仅是简单的线性项时。另一个常见错误是在与乘积或商法则同时使用时误用链式法则。务必明确识别复合的“层次”：如果需要，写出 u 的代入。在化简之前，反复检查是否已经乘上了 du/dx。
Below is a quick reference table for standard derivatives using the chain rule:
下表是利用链式法则的标准导数速查表：
| Function f(x) | Derivative f'(x) |
|---|---|
| [f(x)]n | n [f(x)]n-1 f'(x) |
| sin(f(x)) | f'(x) cos(f(x)) |
| cos(f(x)) | -f'(x) sin(f(x)) |
| e^(f(x)) | f'(x) e^(f(x)) |
| ln(f(x)) | f'(x)/f(x) |
12. Exam Tips for Edexcel A-Level | Edexcel A-Level 考试技巧
In Edexcel exams, chain rule questions often appear in the context of finding stationary points, optimisation, or connected rates of change. Show all steps clearly, especially when introducing a substitution. When asked to prove a derivative, use the chain rule with precise Leibniz notation to gain full marks. Practise recognising when to apply the chain rule within larger problems, such as in differential equations or when finding gradients of parametric curves.
在 Edexcel 考试中，链式法则的题目常常出现在求驻点、优化问题或相关变化率的情境中。清晰展示所有步骤，尤其是在引入代换时。当要求证明某个导数时，使用精确的莱布尼茨记法书写链式法则以获得满分。练习在更复杂的问题中识别何时应用链式法则，例如在微分方程中或求参数曲线的梯度时。
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