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The given function is:

\[
f(x) = \sqrt{4x^2}
\]

First, let's simplify the expression:

\[
f(x) = \sqrt{4x^2} = |2x|
\]

Now, since \( f(x) = |2x| \), we need to take the derivative considering the absolute value function.

The derivative of \( |2x| \) is given by:

\[
f'(x) =
\begin{cases}
2, & \text{if } x > 0 \\
-2, & \text{if } x < 0 \\
\text{undefined}, & \text{if } x = 0
\end{cases}
\]

Therefore, the first derivative of the function is piecewise:

\[
f'(x) =
\begin{cases}
2, & \text{if } x > 0 \\
-2, & \text{if } x < 0
\end{cases}
\]

And at \(x = 0\), the derivative is undefined because of the non-differentiability of the absolute value function at that point.

Would you like more details or have any questions?

Here are five follow-up questions:
1. How do we handle derivatives of absolute value functions in general?
2. Why is the derivative undefined at \(x = 0\)?
3. How can we find the second derivative of \(f(x)\)?
4. What happens if \(f(x)\) was defined differently for negative \(x\)?
5. How does the derivative change if the function was \( f(x) = \sqrt{4x^4} \) instead?

**Tip:** When taking derivatives involving absolute values, always consider the piecewise definition of the function.

Explore how to calculate the first derivative of f(x) = √(4x²), simplified to |2x|, using piecewise differentiation. Understand key concepts in calculus such as the handling of absolute value functions and non-differentiability at specific points. Ideal for students studying high school calculus.

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