How does a function's smoothness relate to the smoothness of its inverse?

Question

Given $f$ is invertable, a classic example of a smooth function having a non-smooth inverse is $f(x) = x^3$ since $x^{1/3}$ is non-differentiable at 0. The converse of this is that $ f(x) = x^{1/3} $ is a non-smooth function whose inverse is smooth. Is there a general formula that, given the smoothness of $f(x)$ on the interval $[a,b]$, states what the smoothness of $f^{-1}(x)$ is on the interval $[f(a),f(b)]$?