MindMap Gallery 7.3 Chain rule of composite functions and derivation of implicit functions
This is a mind map about the chain rule of 7.3 composite functions and the derivation of implicit functions. It is necessary to share the review materials for review and preview to improve learning efficiency. I hope it will be helpful to everyone in preparing for the exam.
Edited at 2024-04-10 09:25:34This is a panoramic infographic—currently sweeping across the web—illustrating the comprehensive applications of OpenClaw, a popular open-source AI agent platform. It systematically introduces this intelligent agent framework—affectionately dubbed "Lobster Farming"—helping readers quickly grasp its core value, technical features, application scenarios, and security protocols. It serves as an excellent introductory guide and practical manual.
這是一張最近風靡全網關於熱門開源AI代理平台OpenClaw的全網應用全景圖解。它系統性地介紹了這款被稱為「養龍蝦」的智慧體框架,幫助讀者快速理解其核心價值、技術特性、應用場景及安全規範,是一份極佳的入門指南與實操手冊。此圖主要針對希望利用AI建構自動化工作流程的技術從業人員、中小企業主及效率追求者,透過9大模組層層遞進,全面剖析了OpenClaw從概念到落地的整個過程。 圖中核心內容首先釐清了「養龍蝦」指涉的是OpenClawd開源智能體,並強調其本質是「AI基建」而非一般聊天機器人。隨後詳細比較其與傳統AI助理的區別,擁有記憶管理、權限控制、會話隔離和異常恢復四大基礎能力,支援跨平台存取和多模型相容(如GPT、Claude、Ollama)。同時,圖解提供了完整的部署方案(雲端/本地/Docker),並列舉了辦公室自動化、內容創作、資料收集等五大應用程式場景。此外,還展示了其火爆程度、政府與大廠佈局、安全部署建議及適合/不適合的人群分類。幫助你快速掌握OpenClaw技術架構與應用價值,指導個人或企業建構AI自動化系統,規避資料外洩與權限失控風險,是學習「執行式AI」轉型的權威參考圖譜。
本圖由萬興腦圖繪製,是針對IT研發崗位的結構化個人履歷模板,完整涵蓋求職核心資訊模組。基本資訊區包含姓名、電話、信箱、求職意願及GitHub連結;專業概要要求以2-3句提煉核心優勢;工作經驗以「公司A高級Java開發工程師」為例,以「透過(行動),達成(量化成果)」格式呈現微服務架構設計、系統效能優化、團隊技術規範制定等職責,公司B經歷則聚焦功能模組開發與Elasticsearch搜尋優化;技能專長分程式語言、後端框架、中介軟體、資料庫、容器雲等維度,清楚展示技術堆疊;專案成果以「電商平台秒殺系統」為例,說明技術棧、架構設計、個人貢獻(Redis Lua庫存原子扣減)及KPI;教育背景包含一流大學電腦專業學歷,以及AWS認證解決方案架構師、軟考中級軟體設計師證書。模板邏輯嚴謹,涵蓋IT研發求職全流程關鍵訊息,幫助求職者清晰、量化展示專業能力。
This is a panoramic infographic—currently sweeping across the web—illustrating the comprehensive applications of OpenClaw, a popular open-source AI agent platform. It systematically introduces this intelligent agent framework—affectionately dubbed "Lobster Farming"—helping readers quickly grasp its core value, technical features, application scenarios, and security protocols. It serves as an excellent introductory guide and practical manual.
這是一張最近風靡全網關於熱門開源AI代理平台OpenClaw的全網應用全景圖解。它系統性地介紹了這款被稱為「養龍蝦」的智慧體框架,幫助讀者快速理解其核心價值、技術特性、應用場景及安全規範,是一份極佳的入門指南與實操手冊。此圖主要針對希望利用AI建構自動化工作流程的技術從業人員、中小企業主及效率追求者,透過9大模組層層遞進,全面剖析了OpenClaw從概念到落地的整個過程。 圖中核心內容首先釐清了「養龍蝦」指涉的是OpenClawd開源智能體,並強調其本質是「AI基建」而非一般聊天機器人。隨後詳細比較其與傳統AI助理的區別,擁有記憶管理、權限控制、會話隔離和異常恢復四大基礎能力,支援跨平台存取和多模型相容(如GPT、Claude、Ollama)。同時,圖解提供了完整的部署方案(雲端/本地/Docker),並列舉了辦公室自動化、內容創作、資料收集等五大應用程式場景。此外,還展示了其火爆程度、政府與大廠佈局、安全部署建議及適合/不適合的人群分類。幫助你快速掌握OpenClaw技術架構與應用價值,指導個人或企業建構AI自動化系統,規避資料外洩與權限失控風險,是學習「執行式AI」轉型的權威參考圖譜。
本圖由萬興腦圖繪製,是針對IT研發崗位的結構化個人履歷模板,完整涵蓋求職核心資訊模組。基本資訊區包含姓名、電話、信箱、求職意願及GitHub連結;專業概要要求以2-3句提煉核心優勢;工作經驗以「公司A高級Java開發工程師」為例,以「透過(行動),達成(量化成果)」格式呈現微服務架構設計、系統效能優化、團隊技術規範制定等職責,公司B經歷則聚焦功能模組開發與Elasticsearch搜尋優化;技能專長分程式語言、後端框架、中介軟體、資料庫、容器雲等維度,清楚展示技術堆疊;專案成果以「電商平台秒殺系統」為例,說明技術棧、架構設計、個人貢獻(Redis Lua庫存原子扣減)及KPI;教育背景包含一流大學電腦專業學歷,以及AWS認證解決方案架構師、軟考中級軟體設計師證書。模板邏輯嚴謹,涵蓋IT研發求職全流程關鍵訊息,幫助求職者清晰、量化展示專業能力。
7.3 Chain rule of composite functions and derivation of implicit functions
Derivative Rules for Composite Functions
chain rule
Variations (essentially tree-like)
Changeable
many
constitute
one
total derivative without bifurcation
Features
are all functions of t
Increasing circumstances
More varied
many
many
One becomes more
one
many
Miscellaneous
This can be solved by drawing a tree diagram
example
Let each item go through the program of the previous step again, find u, y, and derive the derivative again. At the same time, it also satisfies the basic operation of the derivative.
The items in the yellow part are traversed again (x y z) and (xyz)
subtopic
But please note that the footer uses the traversal method
Total differential form invariance (differentiation method)
Suitable for composite binary functions
Differential method to find partial derivatives
First find the partial derivative of z with respect to u and v, and write dz (the essence of differential calculus)
Expand
d(xy)=ydxxdy
d(x y)=dx dy
get
According to the total differential definition, we get
Of course, you can also use the chain derivation rule to calculate
The method is to create a tree diagram
Derivation of implicit functions
F(x, y) = case of type 0 equation
First derivative (mainly for binary functions)
formula
Derivation Let y=f(x)
theorem
condition
has continuous partial derivatives
main body
in conclusion
Set a certain neighborhood
Derivative formula
If you want to find the second derivative, you need to find the derivative of Fx/Fy.
method
example
First use the implicit function theorem
Differentiate on both sides
In order to get dy/dx
formula method
Find the derivative of both sides
You can also proceed with the derivation to find the second derivative
There is an implicit function at (0, 0)
beg
F(x, y, z) = case of type 0 equation
theorem
condition
in conclusion
formula
example
Ask for dz
Fayi
Method 2
Differentiate on both sides
Fayi
Method 2
Differentiate on both sides
Mainly master the methods of differentials on both sides
x y z
dx dy dz
xyz
xydz yzdx xzdy
The case of the system of equations
Just do the math and you'll know
The denominator is the determinant composed of the coefficients before x and y on the left
To find x, replace the coefficient of the column of x with the column on the right side of the equation (c1, c2)
The solution is as above
F,G
Jacobian
example
Method 1
Find the partial derivatives of x on both sides, and then use the method of solving quadratic equations to find the first two
Required conditions
Just find the solution
Then use both sides to find the partial derivative of y, and use the same method as above to find
Method 2
Sometimes implicit functions may be converted into explicit functions, and these methods may not be used.
Find partial derivatives of explicit functions directly
Differentiate on both sides
Differentiate both sides
Obviously, the four things you are looking for can be seen in just one sentence.
shortcoming
Large amount of calculation
Solution
First, according to the deformation of the mother formula, express v and u as x and y, so direct differentiation is also possible.
advantage
Intuitive