Definition

Definition

For differentiable functions and , has derivative

Since multi-variable derivative are matrices, the above multiplication is matrix multiplication.

For example, for and , we have

Tree Diagram

The tree diagram (some people also call it a dependency graph) is an alternative visual tool to help construct chain rule.

In a tree diagram, the dependent variable sit on top, and independent variables sit on bottom, with intermediate variables sit in the middle.

For example, to construct chain rule for a function where each is depending on two variables and , we can have the following diagram:

Chain Rule and Coordinate Transformation

One application of the chain rule is to transform the coordinate system. When a problem has circular symmetry for instance, it doesn’t make sense to write in Cartesian coordinates; the problem is much easier to solve in polar coordinates.

  • total differential - A lot of problems can be solved by either chain rule or expanding the total differential, though using the chain rule is usually the easier way