Answer is: option4
\( e \)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:
\[ (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))} \]
Step 2: Solve for \( f^{-1}(2) \)
We need to find \( x \) such that \( f(x) = 2 \). The equation \( f(x) = 1 + \ln x \) gives us:
\[ 1 + \ln x = 2 \implies \ln x = 1 \]
Solving this for \( x \) gives:
\[ x = e \]
Thus, \( f^{-1}(2) = e \).
Step 3: Find \( f'(x) \)
The derivative of \( f(x) = 1 + \ln x \) is:
\[ f'(x) = \frac{1}{x} \]
Step 4: Evaluate \( f'(e) \)
Substitute \( x = e \) into \( f'(x) \):
\[ f'(e) = \frac{1}{e} \]
Step 5: Apply the inverse function derivative formula
Now, apply the inverse function derivative formula:
\[ (f^{-1})'(2) = \frac{1}{f'(e)} = \frac{1}{\frac{1}{e}} = e \]
Final Answer:
The value of \( (f^{-1})'(2) \) is \( e \), so the correct answer is (D).