# Reciprocal derivative

The reciprocal difference quotient is

or

The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as x1 and x2 approach x:

or as h approaches zero:

The reciprocal derivative of a linear function, f(x) = ax + b, is

The reciprocal derivative of a power function, f(x) = xn, is

In general the reciprocal derivative is the reciprocal of the ordinary derivative:

by the inverse function theorem.

The reciprocal derivative of a constant times a function is the reciprocal constant times the reciprocal derivative of the function:

The reciprocal derivative of the sum of two functions is the reciprocal sum of the two reciprocal derivatives:

in which the squared plus represents reciprocal addition. One may easily prove the following for reciprocally differentiable functions f and g:

If the functions f and g and inverses of one another, then by the inverse-function theorem:

Why would someone use the reciprocal derivative instead of the derivative? The reason is that they might rather work with a function than with the inverse function. In the neighborhood of a non-zero derivative according to the inverse-function theorem, the reciprocal of a derivative equals the derivative of the inverse function (assuming non-zero derivatives). So one can get at the inverse function through the reciprocal derivative.

This applies to instantaneous rates of change such as (time) speed, dx(t)/dt. These rates are appropriate if time is the independent variable, but if distance is the independent variable, then the converse rate, dt(x)/dx, would be appropriate. But we are accustomed to working with speed rather than its converse, pace. So instead, we can use the inverse pace or space speed, (dt(x)/dx)-1dx/dt(x), which is the reciprocal derivative of t(x) rather than the derivative of t(x).

References: “Properties of Reciprocal Derivatives” by M. M. Pahirya and R. A. Katsala, Ukrainian Mathematical Journal, Vol. 62, No. 5, 2010, pages 816-823. “Vorlesungen über Differenzenrechnung” by Niels Erik Nørlund. Springer, Berlin 1924, pages 415, 426.