1. State the problem: Find the derivative $\frac{dR}{dq}$ where $R = q \sqrt{(1000 - q)^2}$.\n\n2. Simplify the expression inside the square root: Since $(1000 - q)^2$ is always non-negative, the square root reduces to the absolute value $|1000 - q|$. Assuming $q < 1000$, $\sqrt{(1000 - q)^2} = 1000 - q$. So, $R = q (1000 - q)$.\n\n3. Rewrite $R$: $R = 1000q - q^2$.\n\n4. Differentiate $R$ with respect to $q$: Using basic power rule, $\frac{dR}{dq} = 1000 - 2q$.\n\n5. Final answer: $$\frac{dR}{dq} = 1000 - 2q$$