Problem
The table below gives selected values for the differentiable and increasing function $f$ and its derivative. If $g(x)=f^{-1}(x)$, what is the value of $g^{\prime}(3)$ ? \begin{tabular}{|c|c|c|} \hline$x$ & $f(x)$ & $f^{\prime}(x)$ \\ \hline-2 & -5 & 9 \\ \hline 1 & -1 & 10 \\ \hline 3 & 2 & 7 \\ \hline 6 & 3 & 6 \\ \hline 8 & 8 & 2 \\ \hline 10 & 9 & 3 \\ \hline \end{tabular}
Solution
Step 1 ：Find the value of f(3) from the table: f(3) = 2
Step 2 ：Find the value of f'(3) from the table: f'(3) = 7
Step 3 ：Find the value of g(3) from the table: g(3) = 6
Step 4 ：Use the formula g'(x) = 1 / f'(g(x))
Step 5 ：Substitute g(3) = 6 and f'(3) = 7 into the formula: g'(3) = 1 / f'(g(3)) = 1 / f'(6) = 1 / 6
Step 6 ：The value of g'(3) is \(\boxed{\frac{1}{6}}\)
