What is the derivation of the multivariable chain rule using the definition of a derivative (and limits) for the simple case of [math] f(x (t), y(t)) [/math] with respect to t? -
![An Alternative Proof of the chain rule and definition of differentiability for multivariable functions | PPT An Alternative Proof of the chain rule and definition of differentiability for multivariable functions | PPT](https://image.slidesharecdn.com/chainrule-131113091337-phpapp01/85/an-alternative-proof-of-the-chain-rule-and-definition-of-differentiability-for-multivariable-functions-2-320.jpg?cb=1669283020)
An Alternative Proof of the chain rule and definition of differentiability for multivariable functions | PPT
![Session 34: The Chain Rule with More Variables 3 | Multivariable Calculus | Mathematics | MIT OpenCourseWare Session 34: The Chain Rule with More Variables 3 | Multivariable Calculus | Mathematics | MIT OpenCourseWare](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/d205755e432d2e8a0bc5f0aa144e633a_MIT18_02SC_L11Brds_15.png)
Session 34: The Chain Rule with More Variables 3 | Multivariable Calculus | Mathematics | MIT OpenCourseWare
![14.5 The Chain Rule - Lecture notes 14.5 - WebAssign 1 The Chain Rule Is a method for calculating - Studocu 14.5 The Chain Rule - Lecture notes 14.5 - WebAssign 1 The Chain Rule Is a method for calculating - Studocu](https://d20ohkaloyme4g.cloudfront.net/img/document_thumbnails/2e6fc32da3c2e0a5ff0fe8cbacae227c/thumb_1200_1553.png)
14.5 The Chain Rule - Lecture notes 14.5 - WebAssign 1 The Chain Rule Is a method for calculating - Studocu
![Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare](https://mitocw.ups.edu.ec/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-34-the-chain-rule-with-more-variables/MIT18_02SC_L11Brds_13.png)
Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare
![Session 32: Total Differentials and the Chain Rule 5 | Multivariable Calculus | Mathematics | MIT OpenCourseWare Session 32: Total Differentials and the Chain Rule 5 | Multivariable Calculus | Mathematics | MIT OpenCourseWare](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/30f0a36c8cac4192de97cddf3e376464_MIT18_02SC_L11Brds_5.png)
Session 32: Total Differentials and the Chain Rule 5 | Multivariable Calculus | Mathematics | MIT OpenCourseWare
![multivariable calculus - I can't understand the chain rule in this question. - Mathematics Stack Exchange multivariable calculus - I can't understand the chain rule in this question. - Mathematics Stack Exchange](https://i.stack.imgur.com/0rXLs.jpg)