1. The problem is to find the derivative of the function $$q = (2r - r^2)^{1/3}$$ with respect to $$r$$. 2. Let $$u = 2r - r^2$$, so the function becomes $$q = u^{1/3}$$. 3. Using the chain rule, $$\frac{dq}{dr} = \frac{dq}{du} \cdot \frac{du}{dr}$$. 4. Compute $$\frac{dq}{du}$$: Since $$q = u^{1/3}$$, $$\frac{dq}{du} = \frac{1}{3} u^{-2/3}$$. 5. Compute $$\frac{du}{dr}$$: Since $$u = 2r - r^2$$, $$\frac{du}{dr} = 2 - 2r$$. 6. Substitute back: $$\frac{dq}{dr} = \frac{1}{3} (2r - r^2)^{-2/3} (2 - 2r)$$. 7. Factor the derivative: $$\frac{dq}{dr} = \frac{1}{3} (2r - r^2)^{-2/3} \cdot 2(1 - r) = \frac{2(1 - r)}{3 (2r - r^2)^{2/3}}$$. Final answer: $$\boxed{\frac{dq}{dr} = \frac{2(1 - r)}{3 (2r - r^2)^{2/3}}}$$