Rates of change are of two kinds. An ordinary rate for the change of *f* relative to a unit of *x* is defined as:

The reciprocal rate is the reciprocal of an ordinary rate with a change of *g* relative to a unit of *x* is defined as:

An ordinary rate has its independent variable in the denominator. The independent variable of a reciprocal rate is in the numerator. A reciprocal rate is the reciprocal of the inverse (or reverse) rate, which has the independent and dependent variables interchanged:

Ordinary rates are added arithmetically:

whereas reciprocal rates are added reciprocally (see post on *reciprocal arithmetic*):

Differential rates are also of two types. The derivative of *f* relative to *x* is defined as:

The reciprocal derivative of *g* relative to *x* is defined as:

This is similar to an ordinary derivative, but it is functionally related to the inverse function by the inverse function theorem.

Ordinary derivatives are added arithmetically:

whereas reciprocal derivatives are added reciprocally:

Ordinary derivatives are the rate of change of a function at a point. Reciprocal derivatives are the rate of change of the inverse function.