To differentiate the function $2\sin^2(x)$, we'll apply the chain rule.

Given:
$f(x) = 2\sin^2(x)$

Step-by-step Solution:

1. Recognize the function structure:
   We can treat $2\sin^2(x)$ as $2 \times (\sin(x))^2$.

2. Differentiate using the chain rule:
   The chain rule states that if we have a function $f(g(x))$, then the derivative is $f'(g(x)) \cdot g'(x)$. Here, $g(x) = \sin(x)$, and $f(u) = u^2$, so the derivative of $u^2$ is $2u$.

   Applying this to $2(\sin(x))^2$:

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

   Simplifying:

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

Thus, the derivative of $2\sin^2(x)$ is:

$\boxed{4\sin(x)\cos(x)}$

Would you like any further explanations or details? Feel free to ask!

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Related Questions:

1. How do you differentiate trigonometric functions like $\sin(x)$ and $\cos(x)$?
2. What is the product rule, and how is it applied in differentiation?
3. Can you explain the chain rule in more detail?
4. How would you differentiate a function like $3\cos^2(x)$?
5. What is the double-angle identity for sine and how can it be derived?

Tip: When dealing with trigonometric identities, sometimes simplifying the expression first (like using the double-angle identity for sine) can make differentiation easier.