CHAIN RULE

Remarks on use of the chain rule.

In general, when three variables x, y and z are connected by a chain of functional relations


then z depends on x. The dependence can be symbolized by

Suppose the original functions y(x) and z(y) are differentiable. Then the chain rule says that z(x) is also differentiable, and

In the application here, for mathematical purposes we regard v as depending on t through the chain of functions


then we have

What is doable is not always useful. Here it is productive to introduce y as the new independent variable because t, the original independent variable, does not appear in equation (1.2) except in the derivative, dv/dt.

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