# Two kinds of vector rates

This post builds on the previous one here.

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

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

The reciprocal vector is defined as follows:

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

Ordinary vector rates are added arithmetically:

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

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

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

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

Ordinary vector derivatives are added arithmetically:

whereas reciprocal vector derivatives are added reciprocally:

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