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 \(\partial z/\partial x = 2xy\), \(\partial z/\partial y = x²\), and \(dx/dt = \cos t\), \(dy/dt = eᵗ\). Substituting gives \(dz/dt = 2xy\cos t + x²eᵗ\), and replacing \(x\) and \(y\) gives the expression entirely in \(t\).

设 \(z = x²y\)，\(x = \sin t\)，\(y = eᵗ\)。则 \(\partial z/\partial x = 2xy\)，\(\partial z/\partial y = x²\)，且 \(dx/dt = \cos t\)，\(dy/dt = eᵗ\)。代入得 \(dz/dt = 2xy\cos t + x²eᵗ\)，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 \(\partial z/\partial s\) and \(\partial z/\partial t\). 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 \(\partial z/\partial s\) 与 \(\partial z/\partial t\)。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 \(\partial F/\partial x = 2x\), \(\partial F/\partial y = 2y\), so \(dy/dx = −2x/(2y) = −x/y\). This matches the slope of the tangent to the circle at any point \((x, y)\) with \(y \neq 0\).

对于圆 \(x² + y² = 25\)，令 \(F(x, y) = x² + y² − 25\)。则 \(\partial F/\partial x = 2x\)，\(\partial F/\partial y = 2y\)，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 \(dz = (\partial z/\partial x)dx + (\partial z/\partial y)dy\). 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 \(dz = (\partial z/\partial x)dx + (\partial z/\partial y)dy\)。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 \(V = \pi r²h\) 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 \(V = \pi r²h\) 的变化率可通过对时间求导得到：

dV/dt = 2\pi r h (dr/dt) + \pi 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 \(\nabla f\) and \(\nabla g\) 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 \(\nabla f\) 与 \(\nabla g\) 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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