Answer is: option3
\( \frac{2}{3} \)Solution:
Step 1: Recall the inverse function derivative rule
If \( g \) is the inverse of \( f \), then the derivative of the inverse function at any point is given by the formula:
\[ g'(x) = \frac{1}{f'(g(x))} \]
Step 2: Apply the given values
We are given that \( g(3) = 4 \) and \( f'(4) = \frac{3}{2} \). Thus, from the formula for the derivative of the inverse function, we can substitute these values:
\[ g'(3) = \frac{1}{f'(g(3))} = \frac{1}{f'(4)} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]
Final Answer: The value of \( g'(3) \) is \( \frac{2}{3} \), so the correct answer is (C).