Answer is: option1

\( 3x^2 \)

Solution:

To find the inverse function \( f(x) \), we solve for \( x \) in terms of \( y \):

1. Replace \( g(x) \) with \( y \): \[ y = \sqrt[3]{x - 1} \] 2. Solve for \( x \):

• Cube both sides:

\[ y^3 = x - 1 \]

• Add 1 to both sides:

\[ x = y^3 + 1 \] 3. Swap \( x \) and \( y \) to express \( f(x) \): \[ f(x) = x^3 + 1 \]

Now, differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx} (x^3 + 1) = 3x^2 \] Final Answer: \[ \boxed{3x^2} \]