1. **Problem 1a:** Find the derivative of $y = 5x^5$. Formula: For $y = ax^n$, the derivative is $y' = a n x^{n-1}$. Step: $y' = 5 \times 5 x^{5-1} = 25x^4$. 2. **Problem 1b:** Find the derivative of $y = 24x^{3.5}$. Step: $y' = 24 \times 3.5 x^{3.5-1} = 84x^{2.5}$. 3. **Problem 1c:** Find the derivative of $y = \frac{1}{x} = x^{-1}$. Step: $y' = -1 x^{-2} = -\frac{1}{x^2}$. 4. **Problem 2a:** Find the derivative of $y = -\frac{4}{x^2} = -4x^{-2}$. Step: $y' = -4 \times (-2) x^{-3} = 8x^{-3} = \frac{8}{x^3}$. 5. **Problem 2b:** Find the derivative of $y = 2x$. Step: $y' = 2$. 6. **Problem 3a:** Find the derivative of $y = 2\sqrt{x} = 2x^{1/2}$. Step: $y' = 2 \times \frac{1}{2} x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}}$. 7. **Problem 3b:** Find the derivative of $y = 3 \sqrt[3]{x^5} = 3x^{5/3}$. Step: $y' = 3 \times \frac{5}{3} x^{5/3 - 1} = 5x^{2/3}$. 8. **Problem 4a:** Find the derivative of $y = \frac{e^2 - e^{-x}}{2}$. Note: $e^2$ is constant, derivative is 0. Step: $y' = \frac{0 - (-e^{-x})}{2} = \frac{e^{-x}}{2}$. 9. **Problem 4b:** Find the derivative of $y = \frac{1 - \sqrt{x}}{x} = \frac{1 - x^{1/2}}{x}$. Rewrite: $y = x^{-1} - x^{-1/2}$. Step: $y' = -1 x^{-2} - (-\frac{1}{2}) x^{-3/2} = -x^{-2} + \frac{1}{2} x^{-3/2} = -\frac{1}{x^2} + \frac{1}{2x^{3/2}}$. **Final answers:** 1a. $25x^4$ 1b. $84x^{2.5}$ 1c. $-\frac{1}{x^2}$ 2a. $\frac{8}{x^3}$ 2b. $2$ 3a. $\frac{1}{\sqrt{x}}$ 3b. $5x^{2/3}$ 4a. $\frac{e^{-x}}{2}$ 4b. $-\frac{1}{x^2} + \frac{1}{2x^{3/2}}$