1. **Stating the problem:** Find the derivative $\frac{dR}{dq}$ where $$R = q\sqrt{(1000 - q)^2}.$$ 2. **Simplify the expression:** Since $\sqrt{(1000 - q)^2} = |1000 - q|$, we rewrite as $$R = q |1000 - q|.$$ 3. **Consider cases for $|1000 - q|$:** - If $q < 1000$, then $|1000 - q| = 1000 - q$. - If $q > 1000$, then $|1000 - q| = q - 1000$. 4. **Case 1 ($q < 1000$):** $$R = q (1000 - q) = 1000q - q^2.$$ Deriving with respect to $q$: $$\frac{dR}{dq} = 1000 - 2q.$$ 5. **Case 2 ($q > 1000$):** $$R = q (q - 1000) = q^2 - 1000q.$$ Deriving with respect to $q$: $$\frac{dR}{dq} = 2q - 1000.$$ 6. **At $q = 1000$:** The function changes behavior, and derivative from left and right are: - From left: $1000 - 2(1000) = 1000 - 2000 = -1000$. - From right: $2(1000) - 1000 = 2000 - 1000 = 1000$. The derivative is not continuous at $q=1000$. **Final answer:** $$\frac{dR}{dq} = \begin{cases} 1000 - 2q & \text{if } q < 1000 \\ 2q - 1000 & \text{if } q > 1000 \end{cases}$$