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:

$\frac{\Delta f(x)}{\bar{\Delta} x}$

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

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

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:

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

Ordinary rates are added arithmetically:

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

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

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

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

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

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

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

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:

$f'_{1}(x)+f'_{2}(x)=\frac{df_{1}}{\bar dx}+ \frac{df_{2}}{\bar dx}=\frac{d(f_{1}+f_{2})}{\bar dx}$

whereas reciprocal derivatives are added reciprocally:

$\grave{}g_{1}(x)+\, \grave{}g_{2}(x)= \left (\frac{dg_{1}}{\bar dx} \right )^{-1} \boxplus \, \left (\frac{dg_{2}}{\bar dx} \right )^{-1}= \left (\frac{d(g_{1}+g_{2})}{\bar dx} \right )^{-1}$

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

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