IB Mathematics: Extensions and Applications of the Chain Rule | IB数学：链式法则的扩展与应用

📚 IB Mathematics: Extensions and Applications of the Chain Rule | IB数学：链式法则的扩展与应用

The chain rule is a fundamental tool in calculus, allowing us to differentiate composite functions. In IB Mathematics, mastery of the chain rule extends far beyond simple polynomial compositions. This article explores its extensions and applications, from multi-step composites to multivariable functions, ensuring you are prepared for both Analysis and Approaches (AA) and Applications and Interpretation (AI) exams.

链式法则是微积分中的基本工具，使我们能够对复合函数求导。在IB数学中，掌握链式法则远不止简单的多项式复合。本文探讨其扩展与应用，从多步复合到多元函数，确保你为分析与方法（AA）以及应用与解释（AI）考试做好准备。

---

1. Review of the Basic Chain Rule | 链式法则基础回顾

The basic chain rule states that if y = f(u) and u = g(x), then the derivative is given by:

基本链式法则指出，若 y = f(u) 且 u = g(x)，则导数为：

dy/dx = dy/du × du/dx = f'(g(x))g'(x)

This rule is the backbone for differentiating any composition of functions.

这一法则是微分任何复合函数的核心。

For IB, you must be able to apply the chain rule quickly. For example, d/dx[(3x² + 1)⁵] = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴.

在IB中，你必须能够快速应用链式法则。例如，d/dx[(3x² + 1)⁵] = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴。

Notice that the derivative of the inner function 3x² + 1 is 6x, which is multiplied by the outer derivative.

注意内函数 3x² + 1 的导数为 6x，它与外层导数相乘。

---

2. Multi-layer Composite Functions | 多层复合函数

Functions often involve more than two layers. For example, y = √(sin(2x)). Here we have three layers: y = √u, u = sin v, v = 2x. The extended chain rule is:

函数通常不止两层。例如，y = √(sin(2x))。这里有三层：y = √u，u = sin v，v = 2x。扩展链式法则为：

dy/dx = dy/du × du/dv × dv/dx

Compute each derivative: dy/du = 1/(2√u), du/dv = cos v, dv/dx = 2. Thus dy/dx = 1/(2√sin(2x)) × cos(2x) × 2 = cos(2x)/√sin(2x).

计算每个导数：dy/du = 1/(2√u)，du/dv = cos v，dv/dx = 2。因此 dy/dx = 1/(2√sin(2x)) × cos(2x) × 2 = cos(2x)/√sin(2x)。

In IB exams, always identify the outermost function first and work inward, one layer at a time.

在IB考试中，始终先识别最外层函数，然后逐层向内运算。

---

3. Chain Rule with Exponential and Logarithmic Functions | 指数与对数函数中的链式法则

The derivatives of e^x and ln x become more interesting when the exponent or argument is a function of x. For example, using the chain rule:

当指数或自变量为 x 的函数时，e^x 和 ln x 的导数变得更有趣。例如，使用链式法则：

d/dx[e^(3x²)] = e^(3x²) · 6x

Similarly, d/dx[ln(sin x)] = 1/(sin x) · cos x = cot x. Remember that the derivative of ln u is u’/u.

类似地，d/dx[ln(sin x)] = 1/(sin x) · cos x = cot x。记住 ln u 的导数是 u’/u。

For a general base a, d/dx[a^(g(x))] = a^(g(x)) ln a · g'(x). For example, d/dx[2^(x³)] = 2^(x³) ln 2 · 3x².

对于一般底数 a，d/dx[a^(g(x))] = a^(g(x)) ln a · g'(x)。例如，d/dx[2^(x³)] = 2^(x³) ln 2 · 3x²。

---

4. Chain Rule in Implicit Differentiation | 隐函数微分中的链式法则

When an equation defines y implicitly in terms of x, we differentiate both sides with respect to x. The chain rule appears because y is a function of x. For example, given x² + y² = 25, differentiating gives 2x + 2y(dy/dx) = 0.

当方程隐式定义 y 为 x 的函数时，我们对两边关于 x 求导。因为 y 是 x 的函数，所以链式法则出现。例如，给定 x² + y² = 25，求导得到 2x + 2y(dy/dx) = 0。

Solving for dy/dx yields:

解出 dy/dx 得到：

dy/dx = -x/y

Notice that the derivative of y² with respect to x is 2y(dy/dx), not simply 2y.

注意 y² 关于 x 的导数是 2y(dy/dx)，而不是简单地 2y。

IB problems often require implicit differentiation for equations like x³ + y³ = 6xy (folium of Descartes), where the chain rule is essential for the y³ and xy terms.

IB题目常要求对类似 x³ + y³ = 6xy（笛卡尔叶形线）的方程进行隐微分，其中链式法则对 y³ 和 xy 项至关重要。

---

5. Chain Rule for Parametric Equations | 参数方程中的链式法则

For a curve defined by x = f(t) and y = g(t), the slope dy/dx is found using the chain rule, provided dx/dt ≠ 0:

对于由 x = f(t)

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）