Chain Rule
From the book
Chapter 11: Matrix Calculus. In the chapter mind map this icon labels Jacobians & Chain Rules. The discussion below is excerpted and lightly edited from § Chain Rule in Mathematics for AI and Machine Learning.
The chain rule is fundamental to computing derivatives of composite functions. When dealing with vector-valued functions, the chain rule takes on a matrix form that depends on the dimensions of the inputs and outputs. The general form for the composition $\mathbf{f}(\mathbf{g}(\mathbf{x}))$ is:
where the matrix dimensions must align correctly for the multiplication to be valid.
What this drawing shows
What you see. Shows how derivatives compose through nested functions and computational graphs.
In the mind map. Chapter 11 — Jacobians & Chain Rules. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 11: Matrix Calculus — see the mind-map node Jacobians & Chain Rules.