Let's differentiate the given function step by step:

The function to differentiate is:

$f(x) = 2\sin(x) + 2\cos^2(x)$

Step 1: Differentiate $2\sin(x)$

The derivative of $\sin(x)$ is $\cos(x)$. So:

$\frac{d}{dx}[2\sin(x)] = 2\cos(x)$

Step 2: Differentiate $2\cos^2(x)$

To differentiate $2\cos^2(x)$, we need to use the chain rule. The derivative of $\cos^2(x)$ is $2\cos(x) \cdot (-\sin(x))$, as we differentiate the square of $\cos(x)$:

$\frac{d}{dx}[2\cos^2(x)] = 2 \cdot 2\cos(x) \cdot (-\sin(x)) = -4\cos(x)\sin(x)$

Final result:

Combining both parts:

$\frac{d}{dx}[2\sin(x) + 2\cos^2(x)] = 2\cos(x) - 4\cos(x)\sin(x)$

This is the derivative of the given function.

Would you like more details or have any further questions on this?

Follow-up questions:

1. How do the chain rule and product rule apply to other trigonometric functions?
2. How does differentiating $\cos^2(x)$ compare to differentiating $\sin^2(x)$?
3. Can you explain how to differentiate higher powers of trigonometric functions like $\cos^3(x)$?
4. What is the importance of simplifying the derivative expression after differentiating?
5. How would you apply this derivative in solving real-world problems?

Tip: Remember that the chain rule is crucial when differentiating composite functions like $\cos^2(x)$.