It seems like there might be a small error in your function notation. Based on what you've provided, I assume you're asking to differentiate the function:

$g(x) = e^{x+2} + 2$

Let's differentiate this with respect to $x$.

Steps:

1. The derivative of $e^{x+2}$ with respect to $x$ is simply $e^{x+2}$, because the exponential function's derivative is the same as the function itself.
2. The derivative of the constant $2$ is $0$, as constants have a zero derivative.

Solution:

$g'(x) = \frac{d}{dx}\left(e^{x+2} + 2\right) = e^{x+2} + 0 = e^{x+2}$

Thus, the derivative is:

$g'(x) = e^{x+2}$

Would you like any further details or have any questions?

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Here are 5 related questions for you to explore:

1. How do you differentiate a function with multiple terms, like polynomials?
2. What is the derivative of a function involving $\ln(x)$?
3. How does the chain rule apply to more complex exponentials, such as $e^{f(x)}$?
4. How would you find the second derivative of $g(x)$?
5. What happens if you differentiate a product of functions?

Tip: When differentiating exponentials, always watch out for whether a chain rule is necessary if the exponent is more complex than a linear function of $x$.