Answer is: option3

0.360

Solution:

Given function: \[ f(x) = 2x + \sin x \]

We need to compute \( g'(2) \), where \( g(x) \) is the inverse function of \( f(x) \).

Using the inverse function derivative formula: \[ g'(x) = \frac{1}{f'(g(x))} \]

Since \( g \) is the inverse of \( f \), we need to solve:

\[ 2x + \sin x = 2 \]

Solving numerically, we get:

\[ x \approx 0.684 \] Compute the derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx} (2x + \sin x) = 2 + \cos x \] Substituting \( x = 0.684 \): \[ f'(0.684) = 2 + \cos(0.684) \] \[ f'(0.684) \approx 2.775 \] Using the inverse function derivative formula: \[ g'(2) = \frac{1}{f'(0.684)} \] \[ g'(2) \approx \frac{1}{2.775} \approx 0.360 \] Final Answer: \[ \boxed{0.360} \]