Find the intervals in which `f(x)=log(2+x)-(2x)/(2+x),\ \ x in R`

Explanation

To find the intervals in which the function \( f(x) = \log(2+x) - \frac{2x}{2+x} \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative \( f'(x) \)

To determine where the function is increasing or decreasing, we first need to compute the derivative \( f'(x) \).

1. The derivative of \( \log(2+x) \) is \( \frac{1}{2+x} \).
2. For the term \( -\frac{2x}{2+x} \), we will use the quotient rule. Let \( u = 2x \) and \( v = 2+x \). Then, the derivative \( \frac{d}{dx}\left(-\frac{u}{v}\right) \) is given by:
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