Day 19: 複合函數的連鎖律 | 90-Day AI Curriculum

01 120 分鐘

複合直覺：導數沿鏈相乘

只有在記錄預測與觀察證據之後才允許求助。

先修條件

Study Days 16-18 的求導規則；如有規則不熟，先開啟當日恢復連結再繼續。

學習目標

對複合函數 y = g(u), u = h(x) 陳述連鎖律 dy/dx = dy/du * du/dx，用微小變化說明其合理性，並對多項式冪應用法則、用展開獨立檢驗。

操作說明

學習來源：在本活動現有的閱讀與練習時段內，參考 MIT OCW 18.01SC Single Variable Calculus 的 Unit 1, Part A, Session 11: Chain Rule; lecture notes and worked example with solution，用自己的話解釋一處與本地任務的聯繫。保留現有練習與驗收檢查；若外部資料無法存取，使用課程自寫的解釋。令 y = (3x + 1)^2。設 u = 3x + 1，則 y = u^2。微小變化 dx 放大為 du = 3 dx，du 再放大為 dy = 2u du，因此 dy/dx = 2u * 3 = 6(3x + 1) = 18x + 6。再先展開驗證：(3x + 1)^2 = 9x^2 + 6x + 1，其導數同樣是 18x + 6。把兩種推導並排寫下，並在計算前預測 x = 1 處的 dy/dx。

練習與預測記錄

先預測再計算：(a) 用連鎖律和展開分別求 d/dx (3x + 1)^2 在 x = 1 處的值；(b) 用連鎖律求 d/dx (2x - 1)^3 在 x = 1 處的值。

標記為已完成

獨立掌握標準

能以命名的內外層函數陳述連鎖律，不看筆記重現 18x + 6 的兩種推導，且預測的 24 和 6 與你親自執行的有限差分結果一致。

權威來源資源與出處

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

MIT OCW 18.01SC Single Variable Calculus

作者/機構: MIT OpenCourseWare (David Jerison) | 原始連結 ↗

驗證版本與日期: Public edition checked 2026-11-01; moving pages may change

選擇理由:

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

來源先修知識:

Functions, algebra and high-school trigonometry.

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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/機構: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始連結 ↗

驗證版本與日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

選擇理由:

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

來源先修知識:

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

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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/機構: Grant Sanderson (3Blue1Brown) | 原始連結 ↗

相關章節/時間戳: Chapter 4 video: composing functions and why derivatives multiply

驗證版本與日期: 2017 video release (verified 2026-10-30)

選擇理由:

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

來源先修知識:

Basic derivative intuition from Study Days 16-18.

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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 分鐘

手算練習與遺漏的內層導數

獨立練習：先預測並保留已記錄的無協助嘗試；之後的任何幫助屬於單獨的「協助式補救」階段，須再通過一次新的同等無協助檢查，該概念才算獨立證據。

先修條件

完成上一項活動；若受阻請使用列出的恢復連結。

學習目標

手算複合冪函數的導數，用多項式展開或有限差分逐一確認，並診斷漏乘內層導數這一經典錯誤。

操作說明

先不看答案。對 f(x) = (5x - 2)^3，連鎖律給出 f'(x) = 3(5x - 2)^2 * 5 = 15(5x - 2)^2。在 x = 1 處用展開驗證：(5x - 2)^3 = 125x^3 - 150x^2 + 60x - 8，其導數 375x^2 - 300x + 60 在 x = 1 處為 135——與 15 * 3^2 相同。再對 g(x) = (x^2 + 1)^2 在 x = 2 處重複，此時內層導數是 2x 而非常數。若在保留的嘗試之後求助，按「協助式補救」處理：原嘗試保持不變，在稱為已補救之前須通過一次新的同等無協助檢查。

練習與預測記錄

手算並驗證：(a) f(x) = (5x - 2)^3 的 f'(1)；(b) g(x) = (x^2 + 1)^2 的 g'(2)。每題同時寫出漏乘內層導數會得到的錯誤答案，並解釋誤差大小。

標記為已完成

獨立掌握標準

在無幫助下給出 135 和 40，各配一段連鎖律推導與獨立檢驗，並能說出常見遺漏丟掉的因子（分別是 5 和 4）。

權威來源資源與出處

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

OpenStax Calculus Volume 1

作者/機構: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始連結 ↗

相關章節/時間戳: Section 3.6 worked examples and end-of-section exercises

驗證版本與日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

選擇理由:

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

來源先修知識:

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

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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/機構: Grant Sanderson (3Blue1Brown) | 原始連結 ↗

相關章節/時間戳: Chapter 4 video: the inner-function correction factor

驗證版本與日期: 2017 video release (verified 2026-10-30)

選擇理由:

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

來源先修知識:

Basic derivative intuition from Study Days 16-18.

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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 分鐘

更深的鏈與有限差分驗證

只有在記錄預測與觀察證據之後才允許求助。

先修條件

完成上一項活動；若受阻請使用列出的恢復連結。

學習目標

對三重嵌套鏈應用連鎖律，並編寫一個用中央有限差分驗證任意手算導數的微型 Python 工具。

操作說明

對 w = (2x^2 + 1)^3，分解為 w = v^3，v = 2x^2 + 1。則 dw/dx = 3v^2 * dv/dx = 3(2x^2 + 1)^2 * 4x = 12x(2x^2 + 1)^2。在 x = 1 處為 12 * 9 = 108，而 w = 27。驗證工具：def central_fd(f, x, h=1e-5): return (f(x + h) - f(x - h)) / (2 * h)。在執行 central_fd(lambda x: (2*x2 + 1)3, 1) 之前先預測輸出值。

練習與預測記錄

寫出 w = (2x^2 + 1)^3 的 dw/dx 三重鏈推導，在 x = 1 處求值，並執行有限差分工具確認。再用該工具複驗 d/dx (3x + 1)^2 在 x = 1 處為 24，記錄任何不一致。

標記為已完成

獨立掌握標準

書面鏈恰好含三個因子且順序正確，預測先於執行，有限差分輸出與 108 和 24 至少四位小數一致。

權威來源資源與出處

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

OpenStax Calculus Volume 1

作者/機構: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始連結 ↗

相關章節/時間戳: Section 3.6: applying the chain rule more than once

驗證版本與日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

選擇理由:

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

來源先修知識:

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

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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/機構: Python Software Foundation | 原始連結 ↗

相關章節/時間戳: Section 3.1.1 Numbers: arithmetic operators used by the finite-difference helper

驗證版本與日期: Python 3.12 documentation (verified 2026-10-30)

選擇理由:

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

來源先修知識:

Able to run a short Python file from the terminal.

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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 分鐘

獨立掌握檢查、折點邊界情形與恢復

獨立掌握檢查（不使用 AI 協助）。

先修條件

完成上一項活動；若受阻請使用列出的恢復連結。

學習目標

在閉卷條件下陳述並應用連鎖律，數值驗證導數，並解釋法則為何在不可微折點失效。

操作說明

檢查期間不看筆記、不用 AI。第 1 部分：陳述 y = g(u), u = h(x) 的連鎖律。第 2 部分：求 d/dx (4x + 3)^2 在 x = 0 處的值並用有限差分驗證。第 3 部分：f(x) = |2x - 2| 在 x = 1 處有折點。計算該點的左差商斜率與右斜率，並解釋為何即使 |u| 和 2x - 2 都是簡單函數，連鎖律在該點仍不適用。

練習與預測記錄

提交：連鎖律陳述、d/dx (4x + 3)^2 在 x = 0 處的值及驗證數值、|2x - 2| 在 x = 1 處的左右斜率，以及一句說明哪個前提不成立。

標記為已完成

獨立掌握標準

無幫助下四部分全部正確：法則陳述、24 且數值一致、斜率 -2 與 +2 並正確解釋前提失效，另附對遺漏內容的誠實恢復記錄。

權威來源資源與出處

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

OpenStax Calculus Volume 1

作者/機構: OpenStax, Rice University (Gilbert Strang, Edwin Herman, et al.) | 原始連結 ↗

驗證版本與日期: OpenStax Calculus Volume 1 web edition (verified 2026-10-30)

選擇理由:

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

來源先修知識:

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

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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/機構: Grant Sanderson (3Blue1Brown) | 原始連結 ↗

驗證版本與日期: 2017 video release (verified 2026-10-30)

選擇理由:

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

來源先修知識:

Basic derivative intuition from Study Days 16-18.

安全回退（離線韌性）:

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

候選來源比較與選擇分析 (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.