Derivative equals inverse

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Here’s kind of a strange problem with an interesting solution: find a function f such that the derivative of f equals the inverse of f for all positive x.

f ′(x) = f−1(x)

This is a differential equation, but a very unusual one, one that cannot be solved using any of the techniques taught in a class on differential equations.

The unique solution is

f(x) = φ(x / φ)φ

where φ is the golden ratio. What an unexpected appearance of the golden ratio!

The problem was proposed by H. L. Nelson and solved by A. C. Hindmarsh. See The American Mathematical Monthly, Vol. 76, No. 6 p. 696.

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John D. Cook, PhD

My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics.

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