1. The problem is to find the derivative $v'$ of the function $v = \sqrt{4x^2 - 1}$. 2. We use the chain rule for derivatives: if $v = (f(x))^{1/2}$, then $v' = \frac{1}{2}(f(x))^{-1/2} \cdot f'(x)$. 3. Here, $f(x) = 4x^2 - 1$, so $f'(x) = 8x$. 4. Applying the chain rule: $$v' = \frac{1}{2}(4x^2 - 1)^{-1/2} \cdot 8x = 4x(4x^2 - 1)^{-1/2}.$$ 5. Your derivative $v' = 4x(4x^2 - 1)^{-1/2}$ is correct!