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