The reciprocal difference quotient is
The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as h approaches zero:
An alternate limit form of the reciprocal derivative definition of f(a) is
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 classical 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 harmonic sum of the two reciprocal derivatives:
in which the circled plus represents harmonic addition. One may easily prove the following for 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 its inverse. 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 speed, dx(t)/dt. These rates are appropriate if time is the independent variable, but if distance is the independent variable, then the inverse rate, dt(x)/dx, would be appropriate. But we are accustomed to working with speed rather than its inverse, pace. In that case we can use the time speed, (dx(t)/dt)-1 = dt(x)/dx, which is the reciprocal derivative of x(t) but the derivative of t(x).
Reference: “Properties of Reciprocal Derivatives” M. M. Pahirya and R. A. Katsala, Ukrainian Mathematical Journal, Vol. 62, No. 5, 2010, pages 816-823.