While doing some calculus problems I came across the remarkable discovery that the function $f(x)=\sqrt{1-x^2}$ actually equals its own derivative at the $x$ value of $-\frac {1}{\phi}$ ($\approx -0.618033$, $\phi$ being the golden ratio, approx. $1.618033$).
This led me to ask, why? I tried setting the two equations equal to each other, and $-1/\phi$ does indeed come out as a solution of the quadratic $\frac {1-\sqrt{5}}{2}$, but what does the square root of $5$ have to do with the golden ratio? And why?
Just for the record, the derivative of f(x) is $\frac {-x}{1-x^2}$ Thanks.
The golden ratio is given by $1:\phi = \phi:(1-\phi)$, which produces the quadratic equation $1-\phi=\phi^2$, i.e., $\phi$ is a solution of $x^2+x-1=0$. The solutions are $\frac{-1\pm\color{red}{\sqrt 5} }{2}$.