On my Calc II quiz today, I couldn't figure out how to do this one problem, and it's driving me nuts just thinking about it... How would you solve it?

f(x)=x^3+3sin(x)+2cos(x)

a=2

Find f^{-1}'(a)

Note that that is the derivative of the inverse function of f. This problem would be easy if I could solve for f^{-1}and plug it into f^{-1}'(x)=1/(f'(f^{-1}(x)), but I couldn't see any way to do that. I racked my brain thinking of a way to implicitly solve for it, but I came up with a blank. Any help?