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