Chain Rule

analysis

Definition

Chain Rule

Let be an open set, a scalar function, and a vector function defined by:

where . Let . Then, the derivative is given by the sum of the products of the partial derivatives along all paths of dependency:

Dependency Graph

The chain rule can be visualised as a tree of dependencies. To find the total derivative of with respect to , one sums the contributions of all possible paths from to .

Simple Scalar Case

For the composition of two scalar functions and , the rule simplifies to: