1. The problem asks to find the derivative of the function $f(x) = 2\sqrt{x}$ at $x=4$. 2. Recall the derivative rule for $\sqrt{x}$: if $f(x) = \sqrt{x} = x^{1/2}$, then $$f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}.$$ 3. Since $f(x) = 2\sqrt{x} = 2x^{1/2}$, by the constant multiple rule, $$f'(x) = 2 \cdot \frac{1}{2} x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}}.$$ 4. Evaluate the derivative at $x=4$: $$f'(4) = \frac{1}{\sqrt{4}} = \frac{1}{2}.$$ 5. Therefore, the value of $f'(4)$ is $\frac{1}{2}$. The correct answer is option c. 1/2.