The derivative of $\sin(2x)$ can be calculated using the chain rule. Here's the step-by-step solution:

1. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$.
2. The chain rule tells us to then multiply by the derivative of the inside function, which in this case is $2x$. The derivative of $2x$ with respect to $x$ is 2.

So, the derivative of $\sin(2x)$ is: $\frac{d}{dx} \sin(2x) = 2 \cos(2x)$

Would you like more details on how the chain rule is applied here, or do you have any other questions?

Here are 5 related questions to explore further:

1. What is the derivative of $\cos(2x)$?
2. How would you differentiate $\sin(3x)$?
3. What is the second derivative of $\sin(2x)$?
4. Can you find the derivative of $\sin^2(2x)$?
5. What is the integral of $\cos(2x)$?

Tip: The chain rule is crucial when differentiating composite functions. Always differentiate the outer function first and then multiply by the derivative of the inner function.