Step-by-step Solution:

We are given the function: \[ f(x) = \tan^{-1} \left(\frac{\sin x}{1 + \cos x}\right) \] We need to find the first derivative \( f'(x) \). \[\] Step 1: Simplify the Argument of \( \tan^{-1} \) \[\] We use the identity: \[ \frac{\sin x}{1 + \cos x} = \tan \frac{x}{2} \] Thus, the function simplifies to: \[ f(x) = \tan^{-1} (\tan \frac{x}{2}) \] Since \( \tan^{-1}(\tan \theta) = \theta \) for principal values: \[ f(x) = \frac{x}{2} \] Step 2: Differentiate \[\] Now, differentiating both sides with respect to \( x \): \[ f'(x) = \frac{d}{dx} \left(\frac{x}{2} \right) \] \[ f'(x) = \frac{1}{2} \] Final Answer \( {\frac{1}{2}} \)