The Chain Rule

A system of interconnected gears provides a concrete physical interpretation of the Chain Rule in calculus. Consider three gears arranged in sequence, where the rotational speeds of the first, second, and third gears are represented by the variables x, z, and y, respectively. The first gear drives the second, and the second drives the third, so the motion of each gear depends on the one preceding it. This structure naturally leads to a two-stage variable relationship that can be analyzed using composite functions.

Composite Functional Relationships

In the first stage, the rotation of the second gear depends on the rotation of the first gear, so the intermediate variable is expressed as z = z(x). In the second stage, the rotation of the third gear depends on the second, resulting in y = y(z). Because the output of the first stage serves as the input to the second, the overall relationship between the first and third gears is a composite function, written as y = y(z(x)).

Rates of Change and the Chain Rule

A small change in the rotational speed of the first gear results in a corresponding change in the second gear, which in turn affects the third gear. To determine the instantaneous rate at which the third gear’s speed changes with respect to the first, the ratio Δy/Δx is considered in the limit as Δx approaches zero. By introducing the intermediate change Δz, this ratio can be expressed as

As Δx approaches 0, the corresponding change Δz also approaches 0, allowing each ratio to be interpreted as a derivative.

This leads directly to the Chain Rule, which states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function:

In the gear system, this result shows that the overall rate of change between the first and third gears is determined by how each individual gear responds to the one driving it.