1. We are given the function $f(x) = -2 \cos x$ and asked to find its derivative $g(x)$ and then evaluate $g\left(\frac{\pi}{2}\right)$. 2. The derivative of $\cos x$ is $-\sin x$, so using the constant multiple rule, the derivative of $f(x)$ is $$g(x) = \frac{d}{dx}[-2 \cos x] = -2 \cdot (-\sin x) = 2 \sin x.$$ 3. Now we evaluate $g\left(\frac{\pi}{2}\right)$ by substituting $x = \frac{\pi}{2}$: $$g\left(\frac{\pi}{2}\right) = 2 \sin \left(\frac{\pi}{2}\right).$$ 4. Since $\sin \left(\frac{\pi}{2}\right) = 1$, we get $$g\left(\frac{\pi}{2}\right) = 2 \times 1 = 2.$$ **Final answer:** $$g\left(\frac{\pi}{2}\right) = 2.$$