We are asked to find the derivative of the function $f(x) = (x+2)^x$ using logarithmic differentiation.

1. Problem Description

We are asked to find the derivative of the function $f(x) = (x+2)^x$ using logarithmic differentiation.

2. Solution Steps

Let $y = (x+2)^x$.

Take the natural logarithm of both sides:

$$\ln y = \ln((x+2)^x)$$

$$\ln y = x \ln(x+2)$$

Now, differentiate both sides with respect to $x$:

$$\frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln(x+2))$$

Using the chain rule on the left side, we get:

$$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \ln(x+2))$$

Using the product rule on the right side:

$$\frac{1}{y} \frac{dy}{dx} = (1)(\ln(x+2)) + x \cdot \frac{1}{x+2} \cdot (1)$$

$$\frac{1}{y} \frac{dy}{dx} = \ln(x+2) + \frac{x}{x+2}$$

Multiply both sides by $y$:

$$\frac{dy}{dx} = y \left( \ln(x+2) + \frac{x}{x+2} \right)$$

Since $y = (x+2)^x$, we substitute $y$ back into the equation:

$$\frac{dy}{dx} = (x+2)^x \left( \frac{x}{x+2} + \ln(x+2) \right)$$

3. Final Answer

The derivative of the function is $(2+x)^x \left( \frac{x}{2+x} + \ln(2+x) \right)$.

Therefore, the answer is A.