Derivative equals inverse
by John from John D. Cook on (#76N9F)
Here's kind of a strange problem with an interesting solution: find a functionf such that the derivative off equals the inverse off 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) = (x / )
where 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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