1. **State the problem:** Find the derivative of the function $$F(p) = V_{max}p - \frac{V_{max}}{P_{max}}p^2$$ with respect to $$p$$. 2. **Recall the derivative rules:** - The derivative of $$ap$$ where $$a$$ is a constant is $$a$$. - The derivative of $$bp^2$$ where $$b$$ is a constant is $$2bp$$. 3. **Apply the derivative:** $$\frac{d}{dp}F(p) = \frac{d}{dp}\left(V_{max}p\right) - \frac{d}{dp}\left(\frac{V_{max}}{P_{max}}p^2\right)$$ 4. **Calculate each term:** $$\frac{d}{dp}\left(V_{max}p\right) = V_{max}$$ $$\frac{d}{dp}\left(\frac{V_{max}}{P_{max}}p^2\right) = 2 \cdot \frac{V_{max}}{P_{max}} p$$ 5. **Combine the results:** $$F'(p) = V_{max} - 2 \frac{V_{max}}{P_{max}} p$$ 6. **Final answer:** $$\boxed{F'(p) = V_{max} - 2 \frac{V_{max}}{P_{max}} p}$$