Answer
\[\left( {{f}^{-1}} \right)'\left( x \right)=\frac{3}{2}\sqrt{x}\]
Work Step by Step
\[\begin{align}
& f\left( x \right)={{x}^{2/3}},x>0 \\
& \text{Write }f\left( x \right)\text{ as }y \\
& y={{x}^{2/3}} \\
& \text{Interchange }x\text{ and }y \\
& x={{y}^{2/3}} \\
& \text{Solve for }y \\
& {{\left( x \right)}^{3/2}}={{\left( {{y}^{2/3}} \right)}^{3/2}} \\
& y={{x}^{3/2}} \\
& \text{Write }y\text{ as }{{f}^{-1}}\left( x \right) \\
& {{f}^{-1}}\left( x \right)={{x}^{3/2}} \\
& \text{Compute the derivative} \\
& \left( {{f}^{-1}} \right)'\left( x \right)=\frac{d}{dx}\left[ {{x}^{3/2}} \right] \\
& \left( {{f}^{-1}} \right)'\left( x \right)=\frac{3}{2}{{x}^{1/2}} \\
& \left( {{f}^{-1}} \right)'\left( x \right)=\frac{3}{2}\sqrt{x} \\
\end{align}\]
