IB Math: Chain Rule and Constrained Extrema | IB数学：链式法则与约束极值

📚 IB Math: Chain Rule and Constrained Extrema | IB数学：链式法则与约束极值

The chain rule is one of the most versatile tools in calculus, enabling us to differentiate composite functions and to track how quantities change along a given path. When extended to several variables, it becomes the bridge between ordinary differentiation and partial differentiation. Meanwhile, constrained extrema problems—where we optimise a function subject to an equation—are elegantly solved using Lagrange multipliers, a method that relies on comparing gradients. In IB Mathematics Analysis and Approaches HL, both techniques frequently appear in Paper 2 and Paper 3, often combined with modelling or geometry contexts. This article offers a complete revision guide: we first consolidate the chain rule from single-variable to multivariable calculus, then apply it to implicit differentiation, and finally develop the theory and practice of constrained optimisation with fully worked examples.

链式法则是微积分中用途最广的工具之一，它帮助我们求复合函数的导数，并能追踪一个量沿某条路径的变化。当把它推广到多变量情形时，它便成为普通导数与偏导数之间的桥梁。另一方面，约束极值问题——即在某个等式限制下求函数的最优值——可以借助拉格朗日乘数法优雅地求解，这一方法的核心在于比较梯度。在IB数学分析与方法高级水平课程中，这两种技巧经常出现在Paper 2和Paper 3中，并常与建模或几何背景结合。本文提供完整的复习指南：先巩固从单变量到多变量的链式法则，再将其应用于隐函数求导，最后系统讲解约束优化的理论与实例。

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1. The Single-Variable Chain Rule | 单变量链式法则

For a composite function y = f(u) where u = g(x), the chain rule states that the derivative of y with respect to x is the product of the derivative of f with respect to u and the derivative of g with respect to x. This rule is fundamental: it allows us to break a complicated function into simpler pieces.

对于复合函数 y = f(u)，其中 u = g(x)，链式法则指出：y 对 x 的导数等于 f 对 u 的导数乘以 g 对 x 的导数。这条法则是基础性的：它让我们把复杂函数拆分成若干简单部分。

dy/dx = dy/du × du/dx

For example, if y = (3x² + 1)⁵, let u = 3x² + 1. Then dy/du = 5u⁴ and du/dx = 6x. Hence dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴. In IB exams, this rule is often tested directly, but more importantly, it is the building block for chain rules with several variables.

例如，若 y = (3x² + 1)⁵，令 u = 3x² + 1，则 dy/du = 5u⁴，du/dx = 6x。因此 dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴。在IB考试中，这条法则常被直接考查，但更重要的是，它是多变量链式法则的基础。

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2. The Multi-Variable Chain Rule | 多元链式法则

When a function z = f(x, y) depends on x and y, and each of these in turn depends on a single parameter t, the total rate of change of z with respect to t is obtained by adding the contributions from x and y. This is the multivariable chain rule.

当一个函数 z = f(x, y) 同时依赖于 x 和 y，而 x 和 y 又都依赖于某个参数 t 时，z 对 t 的总变化率等于来自 x 和 y 的两部分贡献之和。这就是多元链式法则。

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

Suppose z = x²y, x = sin t, and y = eᵗ. Then ∂ z/∂ x = 2xy, ∂ z/∂ y = x², and dx/dt = cos t, dy/dt = eᵗ. Substituting gives dz/dt = 2xycos t + x²eᵗ, and replacing x and y gives the expression entirely in t.

设 z = x²y，x = sin t，y = eᵗ。则 ∂ z/∂ x = 2xy，∂ z/∂ y = x²，且 dx/dt = cos t，dy/dt = eᵗ。代入得 dz/dt = 2xycos t + x²eᵗ，再将 x 和 y 用 t 表示即可。

If x and y depend on two variables, say s and t, we can write a similar formula for ∂ z/∂ s and ∂ z/∂ t. Each partial derivative is a sum of the same form.

如果 x 和 y 同时依赖两个变量，比如 s 和 t，我们可以写出类似的公式求 ∂ z/∂ s 与 ∂ z/∂ t。每个偏导数都是同样形式的和。

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3. Implicit Differentiation via the Chain Rule | 利用链式法则进行隐函数求导

An equation of the form F(x, y) = 0 defines y implicitly as a function of x. Differentiating both sides with respect to x, using the chain rule on the y-term, yields a direct formula for dy/dx.

形如 F(x, y) = 0 的方程把 y 隐式地定义为 x 的函数。对两边关于 x 求导，并在含 y 的项上使用链式法则，便可直接得到 dy/dx 的公式。

dy/dx = −(∂F/∂x)/(∂F/∂y)

For the circle x² + y² = 25, write F(x, y) = x² + y² − 25. Then ∂ F/∂ x = 2x, ∂ F/∂ y = 2y, so dy/dx = −2x/(2y) = −x/y. This matches the slope of the tangent to the circle at any point (x, y) with y ≠ 0.

对于圆 x² + y² = 25，令 F(x, y) = x² + y² − 25。则 ∂ F/∂ x = 2x，∂ F/∂ y = 2y，所以 dy/dx = −2x/(2y) = −x/y。这就是圆上任意非水平点 (x, y) 处切线的斜率。

Implicit differentiation is an application of the chain rule that unifies the single-variable and multivariable perspectives. In constrained optimisation, we often differentiate the constraint equation in a similar way.

隐函数求导是链式法则的一种应用，它把单变量与多变量的视角统一起来。在约束优化中，我们经常要以类似方式对约束方程求导。

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4. Total Differentials and Related Rates | 全微分与相关变化率

The total differential of z = f(x, y) is given by dz = (∂ z/∂ x)dx + (∂ z/∂ y)dy. It approximates the small change in z when x and y change by small amounts. This is the differential form of the chain rule and is useful in error estimation and related-rates problems.

z = f(x, y) 的全微分为 dz = (∂ z/∂ x)dx + (∂ z/∂ y)dy。它近似表示当 x 和 y 各自发生微小变化时 z 的微小变化量。这是链式法则的微分形式，常用于误差估计和相关变化率问题。

For example, if the radius r and height h of a cylinder grow at known rates, the rate at which the volume V = π r²h changes is found by differentiating V with respect to time:

例如，若圆柱的半径 r 与高度 h 以已知速率变化，则体积 V = π r²h 的变化率可通过对时间求导得到：

dV/dt = 2π r h (dr/dt) + π r² (dh/dt)

This is nothing but the multivariable chain rule with t as the independent variable. A clear understanding of total differentials helps you later interpret Lagrange multipliers, where the change in the optimal value is proportional to the change in the constraint.

这正是以 t 为自变量的多元链式法则。透彻理解全微分有助于后面解读拉格朗日乘子——最优值的变化量与约束的变化量成正比。

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5. What Is a Constrained Extremum? | 什么是约束极值？

In many real-world problems, we need to maximise or minimise a function f(x, y) subject to an additional condition g(x, y) = 0. The resulting optimum is called a constrained extremum. For instance, we may maximise area given a fixed perimeter, or minimise distance from a point to a curve.

在许多实际问题中，我们需要在附加条件 g(x, y) = 0 下最大化或最小化函数 f(x, y)。这样得到的最优值称为约束极值。例如，在给定周长下求最大面积，或求某点到一条曲线的最短距离。

Graphically, a constrained optimum occurs where a level curve of f is tangent to the constraint curve g = 0. At such a point, the gradients ∇ f and ∇ g are parallel vectors. This geometric insight leads directly to the method of Lagrange multipliers.

从几何上看，约束最优点出现在 f 的等值线与约束曲线 g = 0 相切处。此时，梯度 ∇ f 与 ∇ g 是平行向量。这一几何洞察直接引出拉格朗日乘数法。

Simple substitution may solve problems where the constraint can be rearranged easily, but substitution often leads to messy algebra or cannot be done at all. Lagrange multipliers provide a systematic alternative that treats all variables symmetrically.

当约束条件容易改写时，简单的代入法可能求解，但代入法往往导致繁琐的代数运算，甚至完全无法执行。拉格朗日乘数法提供了一种系统性的替代方案，对称地处理

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