3 Chain Rules | 三大链式法则

📚 3 Chain Rules | 三大链式法则

In IB Mathematics, the chain rule is far more than just a formula for differentiating composite functions. It is a fundamental tool that appears in three distinct but interconnected forms: the single-variable chain rule, the multivariable chain rule for partial derivatives, and implicit differentiation. Mastering these three applications will give you the confidence to tackle a wide range of problems in calculus, from basic derivatives to related rates and beyond.

在 IB 数学中，链式法则远不只是用来求复合函数导数的一个公式。它是一种基本工具，以三种不同但相互关联的形式出现：一元函数的链式法则、多元函数偏导数的链式法则以及隐函数求导。掌握这三种应用将使你有信心应对微积分中从基础求导到相关变化率等各种问题。

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1. The Single-Variable Chain Rule | 一元链式法则

ivative of y with respect to x is given by dy/dx = dy/du × du/dx. In function notation, (f ∘ g)'(x) = f'(g(x)) · g'(x). This is the classic chain rule that allows you to peel back layers of functions.

如果 y = f(u) 且 u = g(x) 都可导，那么 y 关于 x 的导数为 dy/dx = dy/du × du/dx。用函数记号表示就是 (f ∘ g)'(x) = f'(g(x)) · g'(x)。这就是经典的一元链式法则，让你能由外到内逐层求导。

The idea is simple: differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function. In the IB syllabus, you will use this rule again and again – for polynomials raised to a power, trigonometric functions of linear expressions, exponentials, and logarithms of more complicated arguments.

这个思想很简单：对外层函数关于内层函数求导，再乘以内层函数的导数。在 IB 课程中，你会反复使用这个法则——处理幂次多项式、含有线性表达式的三角函数、指数函数以及对数函数等复合函数。

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2. Example: Basic Chain Rule in Action | 示例：一元链式法则实战

Find the derivative of y = (3x² + 5)⁴.

求 y = (3x² + 5)⁴ 的导数。

· 6x = 24x(3x² + 5)³. You must always remember to multiply by the derivative of the inside – forgetting this step is one of the most common mistakes.

设 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x。应用链式法则，dy/dx = 4u³ · 6x = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³。你必须记住要乘以内层函数的导数——忘记这一步是最常见的错误之一。

), the derivative is cos(cos(x²)) · (−sin(x²)) · 2x. Each layer adds one more derivative factor.

当内部表达式本身又包含复合函数时，你只需要把链式法则延伸下去：如果 y = sin(cos(x²))，导数就是 cos(cos(x²)) · (−sin(x²)) · 2x。每一层都会增加一个导数因子。

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3. Chain Rule with Tables and Recognising Patterns | 表格记法与模式识别

Many IB problems ask you to evaluate a derivative at a specific point using a table of function values. You might be given f(2), f'(2), g(2), g'(2) and asked for (f ∘ g)'(2). The chain rule says it is f'(g(2)) · g'(2). Always substitute carefully and keep track of which derivative you are evaluating.

许多 IB 题目会让你用函数数值表格求某一点的导数值。你可能会看到 f(2)、f'(2)、g(2)、g'(2) 等值，然后要求计算 (f ∘ g)'(2)。链式法则给出 f'(g(2)) · g'(2)。务必仔细代入，弄清楚你正在求的是哪一层导数的值。

This table summarises the basic form and its repeated extension:

下面的表格总结了基本形式及其重复延伸：

Function	Derivative	Notes
f(g(x))	f'(g(x)) g'(x)	One inner function
f(g(h(x)))	f'(g(h(x))) g'(h(x)) h'(x)	Work from outside in

Recognising the pattern of 'derivative of outer, times derivative of inner' helps you apply the rule quickly without substitution every time.

识别“外层导数乘以内层导数”的模式有助于你快速应用法则，无需每次都做代换。

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4. Transition to Multivariable Functions | 向多元函数的过渡

. When x and y themselves depend on a parameter t, the total derivative of z with respect to t requires the multivariable chain rule. This is the second major form of the chain rule.

在 IB 分析与方法 HL 课程中，你还会接触到两个或三个变量的函数，例如 z = f(x, y)。当 x 和 y 自身又依赖于参数 t 时，z 对 t 的全导数就需要用到多元链式法则。这就是链式法则的第二种主要形式。

for the direct dependence of f on x and y, while dx/dt and dy/dt are ordinary derivatives.

如果 z = f(x, y) 且 x = x(t)，y = y(t)，那么 dz/dt = ∂f/∂x · dx/dt + ∂f/∂y · dy/dt。请留意这里对 f 关于 x 和 y 的偏导数使用了偏导符号 ∂，而 dx/dt 和 dy/dt 仍然是普通导数。

unction of t, then df/dt = ∂f/∂x dx/dt + ∂f/∂y dy/dt + ∂f/∂z dz/dt. It is often called the 'total derivative' and appears in optimisation and related rates problems.

这个法则可以自然推广：若 f 依赖于三个变量 x、y、z，而每个变量又都是 t 的函数，则 df/dt = ∂f/∂x dx/dt + ∂f/∂y dy/dt + ∂f/∂z dz/dt。这常被称为“全导数”，出现在最优化和相关变化率问题中。

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5. Multivariable Chain Rule – Tree Diagrams | 多元链式法则——树形图

A powerful way to set up the multivariable chain rule is to draw a dependency tree. Draw the final variable (e.g. z) at the top, branching down to intermediate variables (x and y), and then to the independent variable (t). For each branch, you include a partial or ordinary derivative. Sum over all paths from z to t.

建立多元链式法则的一种高效方法就是画依赖关系树。把最终变量（如 z）放在顶部，向下分支到中间变量（x 和 y），再连到自变量（t）。每条分支上标注一个偏导数或普通导数，然后对所有从 z 到 t 的路径求和。

For example, if z = x² + xy, x = e^t, y = t², then ∂z/∂x = 2x + y, ∂z/∂y = x. So dz/dt = (2x + y) · e^t + x · 2t. Substitute x and y: dz/dt = (2e^t + t²) e^t + e^t · 2t = 2e^2t + t² e^t + 2t e^t.

例如，设 z = x² + xy，x = e^t，y = t²，则 ∂z/∂x = 2x + y，∂z/∂y = x。于是 dz/dt = (2x + y) · e^t + x · 2t。代入 x 和 y 得 dz/dt = (2e^t + t²) e^t + e^t · 2t = 2e^2t + t² e^t + 2t e^t。

miss a pathway. It is invaluable when x and y themselves depend on more than one parameter – = x(u, v) and y = y(u, v), then you would use partial derivatives with respect to u and v.

树形图能确保你不会遗漏任何一条路径。当 x 和 y 又依赖于多个参数时，这种方法尤其宝贵——比如若 x = x(u, v) 且 y = y(u, v)，那么你就要分别对 u 和 v 求偏导数。

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6. Common Pitfalls with Partial Derivatives | 偏导数的常见陷阱

Students often confuse ∂f/∂x with df/dx. In a multivariable context, ∂f/∂x means 'differentiate f with respect to x while treating all other variables as constants' . For the chain rule, you must use the correct type of derivative for each branch. For direct dependence of f on x, use ∂; for the dependence of x on t, use d.

学生经常混淆 ∂f/∂x 和 df/dx。在多元情形下，∂f/∂x 的意思是“关于 x 求导，而把其他所有变量视为常数”。在链式法则中，你必须为每一条分支使用正确的导数类型。对于 f 直接依赖于 x 的部分，用 ∂；对于 x 依赖于 t 的部分，用 d。

Another mistake is to differentiate after substituting. While you can substitute x(t) and y(t) into f to get a single-variable function of t and then differentiate, the multivariable chain rule is often faster and provides insight into how each variable contributes. In IB questions, you are frequently asked to use the chain rule explicitly.

另一个错误是先代入再求导。虽然你可以将 x(t) 和 y(t) 代入 f 得到关于 t 的一元函数再求导，但多元链式法则往往更快，并且能揭示每个变量的贡献。在 IB 题目中，经常会明确要求你使用链式法则。

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7. Implicit Differentiation – The Third Form | 隐函数求导——第三种形式

t given explicitly as a function of x. By differentiating both sides with respect to x and applying the chain rule to terms involving y, you can find dy/dx.

链式法则的第三种核心用法是隐函数求导。许多曲线是由像 x² + y² = 25 或 x³y + y² = 4 这样的方程定义的，其中 y 并没有显式地表示成 x 的函数。通过对等式两边关于 x 求导，并对含 y 的项应用链式法则，你就能求出 dy/dx。

When you see y² in an equation, think of it as a composite function: (y(x))². Differentiating with respect to x gives 2y · dy/dx by the chain rule. Similarly, sin(y) becomes cos(y) · dy/dx, and e^y becomes e^y · dy/dx.

当你在方程中看到 y² 时，要把它视为复合函数：(y(x))²。关于 x 求导，根据链式法则得到 2y · dy/dx。类似地，sin(y) 变成 cos(y) · dy/dx，e^y 变成 e^y · dy/dx。

iable chain rule applied to expressions where y is a function of x. The formula dy/dx = −F_x / F_y ct consequence of the multivariable chain rule, making these three forms deeply connected.

因此，隐函数求导只不过是将一元链式法则应用于 y 为 x 函数的表达式而已。对 F(x, y) = 0，公式 dy/dx = −F_x / F_y 直接来源于多元链式法则，这说明这三种形式是紧密相连的。

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8. Step-by-Step Implicit Differentiation | 隐函数求导步骤

To differentiate x³y + y² = 4 implicitly:

对 x³y + y² = 4 进行隐函数求导：

Step 1. Differentiate both sides with respect to x: d/dx(x³y) + d/dx(y²) = 0.

步骤1：两边关于 x 求导：d/dx(x³y) + d/dx(y²) = 0。

Step 2. Apply the product rule to x³y: 3x² · y + x³ · dy/dx. Apply the chain rule to y²: 2y · dy/dx.

步骤2：对 x³y 应用乘积法则：3x² · y + x³ · dy/dx。对 y² 应用链式法则：2y · dy/dx。

Step 3. Collect the dy/dx terms: x³ dy/dx + 2y dy/dx = −3x²y.

步骤3：合并含 dy/dx 的项：x³ dy/dx + 2y dy/dx = −3x²y。

Step 4. Factor and solve: dy/dx (x³ + 2y) = −3x²y, so dy/dx = −3x²y / (x³ + 2y).

步骤4：提取公因式并求解：dy/dx (x³ + 2y) = −3x²y，因此 dy/dx = −3x²y / (x³ + 2y)。

This method works for any relation provided you differentiate term by term and always append dy/dx when differentiating a function of y.

这种方法适用于任何关系式，只要逐项求导，并在对 y 的函数求导时总是附加上 dy/dx 即可。

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9. Implicit Differentiation with Partial Derivatives | 利用偏导数的隐函数求导

n in the form F(x, y) = 0, the multivariable chain rule gives a clean shortcut: dy/dx = −F_x / F_y, where F_x = ∂F/∂x and F_y a comes from differentiating F(x, y) = 0 with respect to x: ∂F/∂x · 1 + ∂F/∂y · dy/dx = 0.

对于形如 F(x, y) = 0 的关系式，多元链式法则提供了一个简洁的捷径：dy/dx = −F_x / F_y，其中 F_x = ∂F/∂x，F_y = ∂F/∂y。这个公式来源于将 F(x, y) = 0 对 x 求导：∂F/∂x · 1 + ∂F/∂y · dy/dx = 0。

For the circle x² + y² − 25 = 0, F_x = 2x, F_y g the earlier result. This approach is particularly useful when the equation is messy, because you compute the partial derivatives directly without solving for y.

对于圆 x² + y² − 25 = 0，F_x = 2x，F_y = 2y，因此 dy/dx = −2x / 2y = −x / y，与之前的结果一致。当方程比较复杂时，这一方法尤其有用，因为你直接计算偏导数，而无需解出 y。

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10. Related Rates – Chain Rule in Context | 相关变化率——链式法则的情境应用

easing when r = 10, you use dA/dt = dA/dr · dr/dt = 2πr · 3. At r = 10, dA/dt = 60π cm²/s.

相关变化率问题是链式法则的经典应用。如果你知道圆的半径 r 以 3 cm/s 增加，要求当 r = 10 时面积 A = πr² 的增加速度，你就可以利用 dA/dt = dA/dr · dr/dt = 2πr · 3。在 r = 10 时，dA/dt = 60π cm²/s。

e radius. Always write down the chain rule connection first, then substitute known values.

链式法则通过 A 关于 r 的导数，将你知道的变化率（dr/dt）与你想要的变化率（dA/dt）联系起来。同样的原理也适用于三维情况，比如当气球被充气时，你可以将体积 V = (4/3)πr³ 与半径联系起来。务必先写出链式法则的关系式，再代入已知数值。

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11. Chain Rule in Integration – Reverse Recognition | 积分中的链式法则——逆识别

re is a simpler reverse form that IB examiners love: recognising integrals of the type ∫ f'(g(x)) g'(x) dx. Because the derivative of f(g(x)) is f'(g(x)) g'(x), its antiderivative is f(g(x)) + C.

链式法则也为换元积分法奠定了基础，但还有一种更简单的逆向形式，深受 IB 考官青睐：识别形如 ∫ f'(g(x)) g'(x) dx 的积分。因为 f(g(x)) 的导数是 f'(g(x)) g'(x)，所以它的原函数就是 f(g(x)) + C。

Example: ∫ 2x cos(x²) dx. Let f(g) = sin(g) with g(x) = x², then f'(g) = cos(g) and g'(x) = 2x. The integrand matches exactly, so the answer is sin(x²) + C. This 'inverse chain rule' is a time-saver and tests your ability to spot derivatives of inner functions.

例如：∫ 2x cos(x²) dx。令 f(g) = sin(g)，g(x) = x²，则 f'(g) = cos(g)，g'(x) = 2x。被积函数完全匹配，所以答案是 sin(x²) + C。这种“链式法则的逆用”能节省时间，并测试你识别内层函数导数的能力。

Be careful, however: this only works when the inner derivative is exactly present. If the constant factor does not match, you must adjust by a constant multiple.

不过要小心：只有当内层导数完全出现时，这一方法才直接适用。如果常数因子不匹配，你必须通过乘以一个常数来调整。

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12. Summary: Three Forms, One Principle | 总结：三种形式，一个原理

The three chain rules – single-variable, multivariable, and implicit differentiation – all share the same core idea: break down the rate of change into a product of intermediate rates along a dependency chain. Whether you are peeling layers from a composite function, tracking partial derivatives across branches, or handling a hidden relation between x and y, the chain rule is your guide.

这三种链式法则——一元、多元以及隐函数求导——都共享同一个核心思想：将变化率分解为沿依赖链的中间变化率的乘积。无论你是要剥离复合函数的层数、沿着分支追踪偏导数，还是要处理 x 和 y 之间的隐含关系，链式法则都是你的指南。

Practice recognising which form is required: a single function of a function calls for the basic rule; a function of several variables depending on a parameter calls for the multivariable rule; an equation mixing x and y points to implicit differentiation. In IB exams, these often appear mixed together, so a solid grasp of all three is essential for top marks.

练习识别需要使用哪种形式：一个函数的函数对应基本法则；依赖于参数的多元函数对应多元法则；混合 x 和 y 的方程则指向隐函数求导。在 IB 考试中，这些形式经常混合出现，因此扎实掌握全部三种对于取得高分至关重要。

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