John D. Cook • 6/30/2026

Derivative equals inverse

This article explores a unique differential equation: find a function f such that f'(x) = f^{-1}(x) for all positive x. Unlike standard differential equations, this problem cannot be solved by typical techniques. The surprising solution involves the golden ratio φ, with f(x) = φ(x/φ)^φ. The problem was originally posed by H. L. Nelson and solved by A. C. Hindmarsh, appearing in The American Mathematical Monthly. The article highlights the unexpected appearance of the golden ratio in a calculus context.

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#Golden Ratio #Inverse Function #Derivative