Answer
It is an increasing function, so it has an inverse (it is one-to-one.)
$\displaystyle \frac{df^{-1}}{dx}=3$
Work Step by Step
$f(x)=\displaystyle \frac{1}{3}x+\frac{5}{6}$
is a linear function with positive slope (the graph of f rises over the whole domain). It is an increasing function.
By Exercise 49, it is one-to-one and has an inverse.
Theorem 1 provides a way to find a formula for $\displaystyle \frac{df^{-1}}{dx}$
$\displaystyle \frac{d}{dx}[f^{-1}(x)]=\frac{1}{f'[f^{-1}(x)]}$
$f'(x)=\displaystyle \frac{1}{3}$, so $\displaystyle \quad f'[f^{-1}(x)]=\frac{1}{3}\quad$ and
$\displaystyle \frac{d}{dx}[f^{-1}(x)]=\frac{1}{1/3}=3$
