# Two kinds of rates

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

$\frac{\Delta&space;f(x)}{\bar{\Delta}&space;x}$

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

$\left&space;(\frac{\Delta&space;g(x)}{\bar{\Delta}&space;x}&space;\right&space;)^{-1}&space;=&space;\frac{\Delta&space;x}{\Delta&space;g(x)}$

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:

$\frac{\Delta&space;y(x)}{\bar{\Delta}&space;x}&space;\,&space;vs.\,&space;\frac{\Delta&space;x(y)}{\bar{\Delta}&space;y}$

$\frac{\Delta&space;f_{1}}{\bar{\Delta}&space;x}+&space;\frac{\Delta&space;f_{2}}{\bar{\Delta}&space;x}=&space;\frac{\Delta&space;f_{1}+\Delta&space;f_{2}}{\bar{\Delta}&space;x}$

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

$\left&space;(\frac{\Delta&space;g_{1}}{\bar{\Delta}&space;x}&space;\right&space;)^{-1}&space;\boxplus&space;\,&space;\left&space;(\frac{\Delta&space;g_{2}}{\bar{\Delta}&space;x}&space;\right&space;)^{-1}=&space;\left&space;(\frac{\Delta&space;g_{1}+\Delta&space;g_{2}}{\bar{\Delta}&space;x}&space;\right&space;)^{-1}$

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

$f'(x)=\frac{\mathrm{d}&space;f(x)}{\mathrm{d}&space;x}=&space;\lim_{h\rightarrow&space;0}\frac{f(x+h)-f(x)}{h}$

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

$\grave{}g(x)=\left&space;(\frac{\mathrm{d}&space;g(x)}{\mathrm{d}&space;x}&space;\right&space;)^{-1}&space;=\left&space;(\lim_{h\rightarrow&space;0}\frac{g(x+h)-g(x)}{h}&space;\right&space;)^{-1}$

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

$f'_{1}(x)+f'_{2}(x)=\frac{df_{1}}{\bar&space;dx}+&space;\frac{df_{2}}{\bar&space;dx}=\frac{d(f_{1}+f_{2})}{\bar&space;dx}$
$\grave{}g_{1}(x)+\,&space;\grave{}g_{2}(x)=&space;\left&space;(\frac{dg_{1}}{\bar&space;dx}&space;\right&space;)^{-1}&space;\boxplus&space;\,&space;\left&space;(\frac{dg_{2}}{\bar&space;dx}&space;\right&space;)^{-1}=&space;\left&space;(\frac{d(g_{1}+g_{2})}{\bar&space;dx}&space;\right&space;)^{-1}$