Is there a function whose derivative is positive and the derivative of its inverse is negative?

Question

Does there exist some bijective function $f(x)$ such that $f'(x)$ is positive everywhere and $(f^{-1})'(x)$ is negative everywhere?

Answer 1

A function and its inverse function are symmetric in respect to the line $y=x$. So, if the initial function is growing, so would its inverse, just at slower pace.

From any point on the plot of initial function you can draw a normal to the $y=x$ line and find the corresponding point of the inverse function. At that point it will have derivative inverse to the derivative of initial function where it is intersecting the normal.

Since $1/x$ has the same sign as $x$, the bothe will have the same sign.

Answer 2

For the function: $y = e^x ~ | ~0 < x < 1 $, I believe I have achieved the desired effect.