# A function whose inverse is equal to its derivative

Here's a problem: find a function \(f\) whose inverse \(f^{-1}\) is equal to its derivative \(f'\).

## A possible approach

Assume the function \(f\) is a polynomial and can be written as

where \(a\) and \(b\) are constants we wish to determine. We know the derivative \(f'\) is given by

and the inverse \(f^{-1}\) by

and so we obtain the following system of equations:

The second equation looks familiar, let's expand on that. Rewriting gives us

which may be solved using the quadratic formula,

The golden ratio. You know when you see the golden ratio pop up somewhere, things are getting more interesting.

Now let's solve for \(a\), we know \(\frac{1}{b} = b - 1\), so

and

Finally we have,

## Are there more?

If so, how many? Infinite? How can we find them?

## Comments

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