1. **Problem statement:** Given the function $$P = 2Q^3 + 5Q^2 + \frac{2}{Q}$$, find its derivative with respect to $$Q$$, denoted as $$\frac{dP}{dQ}$$. 2. **Rewrite the function for differentiation:** $$P = 2Q^3 + 5Q^2 + 2Q^{-1}$$ 3. **Apply the power rule:** For a term $$aQ^n$$, the derivative is $$anQ^{n-1}$$. So, - Derivative of $$2Q^3$$ is $$2 \times 3 Q^{3-1} = 6Q^2$$. - Derivative of $$5Q^2$$ is $$5 \times 2 Q^{2-1} = 10Q$$. - Derivative of $$2Q^{-1}$$ is $$2 \times (-1) Q^{-1-1} = -2Q^{-2}$$. 4. **Combine derivatives:** $$\frac{dP}{dQ} = 6Q^2 + 10Q - 2Q^{-2}$$ 5. **Compare with provided options:** The correct derivative matches the option: $$6Q^2 + 10Q - 2Q^{-2}$$ **Final answer:** $$6Q^2 + 10Q - 2Q^{-2}$$