iSoul In the beginning is reality.

# Reciprocal derivative, part 1

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:

Reference: “Properties of Reciprocal Derivatives” M. M. Pahirya and R. A. Katsala, Ukrainian Mathematical Journal, Vol. 62, No. 5, 2010, pages 816-823.