f inverse and derivatives of such.

[superevilcube]

\(\displaystyle Find (f^-^1)'(a)\) for the function \(\displaystyle f\) and real number \(\displaystyle a\):

\(\displaystyle f(x)=x^3+2x-1\) , \(\displaystyle a=2\)

I'm not entirely sure what to do. Do I take the derivative of \(\displaystyle f\), plug in the real number \(\displaystyle a\) and then flip the fraction to get the answer?

\(\displaystyle f'(x)=3x^2+2\)
\(\displaystyle f'(2)=3(2)^2+2\)
\(\displaystyle f'(2)=14\)
\(\displaystyle (f^-^1)'(2)=1/14\)

The answer in my book is, however, \(\displaystyle 1/5\). What did I do wrong, or what am I supposed to do?

 

[pka]

Note that \(\displaystyle f(1) = 2\) so \(\displaystyle \left( {f^{ - 1} (2)} \right)' = \frac{1}{{f'(1)}}\)