The Chain Rule
Differentiating Compositions of Functions
The chain rule is the master tool of calculus, enabling the differentiation of composed functions. This chapter derives the multivariate chain rule via local linear approximation, explores matrix-form justification, and covers the key special cases: one underlying variable, two underlying variables, and non-canonical input/output types.
Sections
Motivation for the Chain Rule
8 min · 2 quiz questions
Setup: Composition of Functions
8 min · 2 quiz questions
Univariate Chain Rule
8 min · 2 quiz questions · Interactive diagram
Chain Rule Notation
7 min · 2 quiz questions · Interactive diagram
Justification via LLA
10 min · 2 quiz questions
Error Analysis
8 min · 2 quiz questions
Multivariate Setup
7 min · 1 quiz question · Interactive diagram
The Multivariate Chain Rule
10 min · 2 quiz questions
Matrix Form Justification
8 min · 1 quiz question · Interactive diagram
One Underlying Variable
8 min · 1 quiz question
Example: One Underlying Variable
8 min · 1 quiz question · Interactive diagram
Detailed Justification
8 min · 1 quiz question
Two Underlying Variables
8 min · 2 quiz questions
Multiple Outputs
8 min · 1 quiz question
Non-Canonical Case: Input is a Scalar
8 min · 1 quiz question
Non-Canonical Case: Output is a Scalar
8 min · 2 quiz questions