Unsure of how I would find the inverse of this function

Question

The question states to

Consider $f(x)=x+2x^3+x^5$. Evaluate $(f^{-1})'(4)$.

I know that $f'(x)=1+6x^2+5x^4.$

However, I am not sure if it is even possible to find the inverse function and if so, how would I go about completing the problem?

Answer 1

You don't compute $f^{-1}$. Since $f(1)=4$, $f^{-1}(4)=1$ and therefore$$(f^{-1})'(4)=\frac1{f'\left(f^{-1}(4)\right)}=\frac1{f'(1)}.$$