Calculus · Superchallenge problem

A differentiable function $f$ is defined implicitly (and uniquely, since $t\mapsto t^3+t$ is strictly increasing) by $f(x)^3+f(x)=8x^3+2x$ for all real $x$, and $f$ is strictly increasing. Let $g=f^{-1}$. Find $g'(2)$.