Day 19: 复合函数的链式法则

01 120 分钟

复合直觉：导数沿链相乘

只有在记录预测和观察证据之后才允许求助。

前提条件

Study Days 16-18 的求导规则；如有规则不熟，先打开当天恢复链接再继续。

学习目标

对复合函数 y = g(u), u = h(x) 陈述链式法则 dy/dx = dy/du * du/dx，用微小变化说明其合理性，并对多项式幂应用法则、用展开独立检验。

操作说明

学习来源：在本活动现有的阅读与练习时段内，参考 MIT OCW 18.01SC Single Variable Calculus 的 Unit 1, Part A, Session 11: Chain Rule; lecture notes and worked example with solution，用自己的话解释一处与本地任务的联系。保留现有练习与验收检查；若外部资料无法访问，使用课程自写的解释。令 y = (3x + 1)^2。设 u = 3x + 1，则 y = u^2。微小变化 dx 放大为 du = 3 dx，du 再放大为 dy = 2u du，因此 dy/dx = 2u * 3 = 6(3x + 1) = 18x + 6。再先展开验证：(3x + 1)^2 = 9x^2 + 6x + 1，其导数同样是 18x + 6。把两种推导并排写下，并在计算前预测 x = 1 处的 dy/dx。

练习与预测记录

先预测再计算：(a) 用链式法则和展开分别求 d/dx (3x + 1)^2 在 x = 1 处的值；(b) 用链式法则求 d/dx (2x - 1)^3 在 x = 1 处的值。

标记为已完成

独立掌握标准

能用命名的内外层函数陈述链式法则，不看笔记复现 18x + 6 的两种推导，且预测的 24 和 6 与你亲自运行的有限差分结果一致。

权威源资源与出处

linking CC BY-NC-SA 4.0 noncommercial share-alike

MIT OCW 18.01SC Single Variable Calculus

作者/机构: MIT OpenCourseWare (David Jerison) | 原始链接 ↗

验证版本与日期: Public edition checked 2026-11-01; moving pages may change

选择理由:

Independent-study teaching sequence with lectures, worked recitations, problem sets and solutions for Days 16-19; OpenStax remains the numbered written reference.

源前提知识:

Functions, algebra and high-school trigonometry.

安全回退（离线韧性）:

Use the original localized explanations, exercises and checks already present in this activity; no external account or purchase is required for that fallback.

候选源比较与选择分析 (2)

OpenStax Calculus Volume 1 ↗ compared_alternative [CC BY-NC-SA 4.0]

Stable numbered text and exercises; retained alongside the lecture and recitation sequence.

Essence of Calculus ↗ compared_alternative [Copyright 3Blue1Brown / Grant Sanderson (no general open license)]

Visual motivation without an answer-keyed exercise sequence; retained for intuition.

linking CC BY-NC-SA 4.0 attribution noncommercial share-alike

OpenStax Calculus Volume 1

作者/机构: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始链接 ↗

验证版本与日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

选择理由:

Openly licensed primary treatment of the scalar chain rule with worked examples that match the day's hand calculations.

源前提知识:

Derivative rules for powers and polynomials (Study Days 16-18).

安全回退（离线韧性）:

The lesson contains complete local derivations and worked answers; offline study is unaffected.

候选源比较与选择分析 (3)

MIT OpenCourseWare 18.01SC Single Variable Calculus ↗ compared_selected_for_visual_practice [CC BY-NC-SA 4.0]

Strong exercises and transcripts, but the noncommercial share-alike terms and course navigation make OpenStax the primary stable text.

Useful reference, but rights and redistribution terms are less suitable for embedded recovery material.

SciPy Numerical Differentiation Documentation ↗ compared_selected_for_numerics [BSD-3-Clause]

A primary API reference for numerical derivative step-size behavior; it complements rather than replaces the hand derivation.

Essence of Calculus

作者/机构: Grant Sanderson (3Blue1Brown) | 原始链接 ↗

相关章节/时间戳: Chapter 4 video: composing functions and why derivatives multiply

验证版本与日期: 2017 video release (verified 2026-10-30)

选择理由:

Visual-first treatment of why the derivative of a composition is a product of derivatives; selected for its geometric explanation of the chain rule.

源前提知识:

Basic derivative intuition from Study Days 16-18.

安全回退（离线韧性）:

All visual intuitions are restated as text derivations and finite-difference checks in the local content.

候选源比较与选择分析 (2)

OpenStax Calculus Volume 1 Section 3.6 ↗ compared_selected_for_day19 [CC BY-NC-SA 4.0]

Chosen as the primary written source; 3Blue1Brown supplies the geometric motivation layer.

Khan Academy: Differential Calculus ↗ compared_alternative [CC BY-NC-SA 4.0]

Good drill coverage but less memorable visual intuition for the chain rule's product structure.

02 120 分钟

手算练习与遗漏的内层导数

独立练习：先预测并保留已记录的无协助尝试；之后的任何帮助属于单独的“协助式补救”阶段，须再通过一次新的同等无协助检查，该概念才算独立证据。

前提条件

完成上一项活动；若受阻请使用列出的恢复链接。

学习目标

手算复合幂函数的导数，用多项式展开或有限差分逐一确认，并诊断漏乘内层导数这一经典错误。

操作说明

先不看答案。对 f(x) = (5x - 2)^3，链式法则给出 f'(x) = 3(5x - 2)^2 * 5 = 15(5x - 2)^2。在 x = 1 处用展开验证：(5x - 2)^3 = 125x^3 - 150x^2 + 60x - 8，其导数 375x^2 - 300x + 60 在 x = 1 处为 135——与 15 * 3^2 相同。再对 g(x) = (x^2 + 1)^2 在 x = 2 处重复，此时内层导数是 2x 而非常数。若在保留的尝试之后求助，按“协助式补救”处理：原尝试保持不变，在称为已补救之前须通过一次新的同等无协助检查。

练习与预测记录

手算并验证：(a) f(x) = (5x - 2)^3 的 f'(1)；(b) g(x) = (x^2 + 1)^2 的 g'(2)。每题同时写出漏乘内层导数会得到的错误答案，并解释误差大小。

标记为已完成

独立掌握标准

在无帮助下给出 135 和 40，各配一段链式法则推导与独立检验，并能说出常见遗漏丢掉的因子（分别是 5 和 4）。

权威源资源与出处

linking CC BY-NC-SA 4.0 attribution noncommercial share-alike

OpenStax Calculus Volume 1

作者/机构: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始链接 ↗

相关章节/时间戳: Section 3.6 worked examples and end-of-section exercises

验证版本与日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

选择理由:

Provides additional independent drill problems once the local exercises are mastered.

源前提知识:

Derivative rules for powers and polynomials (Study Days 16-18).

安全回退（离线韧性）:

The exercise answers are fully worked locally, so the drill never depends on the external site.

候选源比较与选择分析 (3)

MIT OpenCourseWare 18.01SC Single Variable Calculus ↗ compared_selected_for_visual_practice [CC BY-NC-SA 4.0]

Strong exercises and transcripts, but the noncommercial share-alike terms and course navigation make OpenStax the primary stable text.

Useful reference, but rights and redistribution terms are less suitable for embedded recovery material.

SciPy Numerical Differentiation Documentation ↗ compared_selected_for_numerics [BSD-3-Clause]

A primary API reference for numerical derivative step-size behavior; it complements rather than replaces the hand derivation.

Essence of Calculus

作者/机构: Grant Sanderson (3Blue1Brown) | 原始链接 ↗

相关章节/时间戳: Chapter 4 video: the inner-function correction factor

验证版本与日期: 2017 video release (verified 2026-10-30)

选择理由:

Explains geometrically why omitting the inner derivative undercounts the change, matching the worked failure case.

源前提知识:

Basic derivative intuition from Study Days 16-18.

安全回退（离线韧性）:

The failure case is demonstrated locally with an expansion check and numerical values.

候选源比较与选择分析 (2)

OpenStax Calculus Volume 1 Section 3.6 ↗ compared_selected_for_day19 [CC BY-NC-SA 4.0]

Chosen as the primary written source; 3Blue1Brown supplies the geometric motivation layer.

Khan Academy: Differential Calculus ↗ compared_alternative [CC BY-NC-SA 4.0]

Good drill coverage but less memorable visual intuition for the chain rule's product structure.

03 120 分钟

更深的链与有限差分验证

只有在记录预测和观察证据之后才允许求助。

前提条件

完成上一项活动；若受阻请使用列出的恢复链接。

学习目标

对三重嵌套链应用链式法则，并编写一个用中心有限差分验证任意手算导数的微型 Python 工具。

操作说明

对 w = (2x^2 + 1)^3，分解为 w = v^3，v = 2x^2 + 1。则 dw/dx = 3v^2 * dv/dx = 3(2x^2 + 1)^2 * 4x = 12x(2x^2 + 1)^2。在 x = 1 处为 12 * 9 = 108，而 w = 27。验证工具：def central_fd(f, x, h=1e-5): return (f(x + h) - f(x - h)) / (2 * h)。在运行 central_fd(lambda x: (2*x2 + 1)3, 1) 之前先预测输出值。

练习与预测记录

写出 w = (2x^2 + 1)^3 的 dw/dx 三重链推导，在 x = 1 处求值，并运行有限差分工具确认。再用该工具复验 d/dx (3x + 1)^2 在 x = 1 处为 24，记录任何不一致。

标记为已完成

独立掌握标准

书面链恰好含三个因子且顺序正确，预测先于运行，有限差分输出与 108 和 24 至少四位小数一致。

权威源资源与出处

linking CC BY-NC-SA 4.0 attribution noncommercial share-alike

OpenStax Calculus Volume 1

作者/机构: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始链接 ↗

相关章节/时间戳: Section 3.6: applying the chain rule more than once

验证版本与日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

选择理由:

Documents repeated application of the chain rule used in the three-link composition drill.

源前提知识:

Derivative rules for powers and polynomials (Study Days 16-18).

安全回退（离线韧性）:

The worked answer derives the three-link product locally step by step.

候选源比较与选择分析 (3)

MIT OpenCourseWare 18.01SC Single Variable Calculus ↗ compared_selected_for_visual_practice [CC BY-NC-SA 4.0]

Strong exercises and transcripts, but the noncommercial share-alike terms and course navigation make OpenStax the primary stable text.

Useful reference, but rights and redistribution terms are less suitable for embedded recovery material.

SciPy Numerical Differentiation Documentation ↗ compared_selected_for_numerics [BSD-3-Clause]

A primary API reference for numerical derivative step-size behavior; it complements rather than replaces the hand derivation.

linking PSF License Version 2

The Python Tutorial: Numbers and Expressions

作者/机构: Python Software Foundation | 原始链接 ↗

相关章节/时间戳: Section 3.1.1 Numbers: arithmetic operators used by the finite-difference helper

验证版本与日期: Python 3.12 documentation (verified 2026-10-30)

选择理由:

Reference for the arithmetic syntax used by the central finite-difference verification helper.

源前提知识:

Able to run a short Python file from the terminal.

安全回退（离线韧性）:

The full helper listing is embedded in the lesson and runs on a standard Python install.

候选源比较与选择分析 (2)

math — Mathematical functions ↗ compared_alternative [PSF License Version 2]

Extends the operator set to transcendental functions; kept as a follow-on once the arithmetic expressions are fluent.

The Python Tutorial: Floating Point Arithmetic Issues ↗ compared_alternative [PSF License Version 2]

Explains binary float representation limits; deferred to the numerics days where tolerance reasoning is rehearsed.

04 120 分钟

独立掌握检查、折点边界情形与恢复

独立掌握检查（不使用 AI 协助）。

前提条件

完成上一项活动；若受阻请使用列出的恢复链接。

学习目标

在闭卷条件下陈述并应用链式法则，数值验证导数，并解释法则为何在不可微折点失效。

操作说明

检查期间不看笔记、不用 AI。第 1 部分：陈述 y = g(u), u = h(x) 的链式法则。第 2 部分：求 d/dx (4x + 3)^2 在 x = 0 处的值并用有限差分验证。第 3 部分：f(x) = |2x - 2| 在 x = 1 处有折点。计算该点的左差商斜率与右斜率，并解释为何即使 |u| 和 2x - 2 都是简单函数，链式法则在该点仍不适用。

练习与预测记录

提交：链式法则陈述、d/dx (4x + 3)^2 在 x = 0 处的值及验证数值、|2x - 2| 在 x = 1 处的左右斜率，以及一句说明哪个前提不成立。

标记为已完成

独立掌握标准

无帮助下四部分全部正确：法则陈述、24 且数值一致、斜率 -2 与 +2 并正确解释前提失效，另附对遗漏内容的诚实恢复记录。

权威源资源与出处

linking CC BY-NC-SA 4.0 attribution noncommercial share-alike

OpenStax Calculus Volume 1

作者/机构: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始链接 ↗

验证版本与日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

选择理由:

States the differentiability hypotheses that explain the kink counterexample in the mastery check.

源前提知识:

Derivative rules for powers and polynomials (Study Days 16-18).

安全回退（离线韧性）:

The edge case is analyzed locally with left and right difference quotients.

候选源比较与选择分析 (3)

MIT OpenCourseWare 18.01SC Single Variable Calculus ↗ compared_selected_for_visual_practice [CC BY-NC-SA 4.0]

Strong exercises and transcripts, but the noncommercial share-alike terms and course navigation make OpenStax the primary stable text.

Useful reference, but rights and redistribution terms are less suitable for embedded recovery material.

SciPy Numerical Differentiation Documentation ↗ compared_selected_for_numerics [BSD-3-Clause]

A primary API reference for numerical derivative step-size behavior; it complements rather than replaces the hand derivation.

Essence of Calculus

作者/机构: Grant Sanderson (3Blue1Brown) | 原始链接 ↗

验证版本与日期: 2017 video release (verified 2026-10-30)

选择理由:

Optional review link for learners who want the geometric picture before the independent check.

源前提知识:

Basic derivative intuition from Study Days 16-18.

安全回退（离线韧性）:

The assessment is fully specified locally and needs no external media.

候选源比较与选择分析 (2)

OpenStax Calculus Volume 1 Section 3.6 ↗ compared_selected_for_day19 [CC BY-NC-SA 4.0]

Chosen as the primary written source; 3Blue1Brown supplies the geometric motivation layer.

Khan Academy: Differential Calculus ↗ compared_alternative [CC BY-NC-SA 4.0]

Good drill coverage but less memorable visual intuition for the chain rule's product structure.