1a. Given $y = 5x^5$, find the derivative $\frac{dy}{dx}$. Using the power rule $\frac{d}{dx} x^n = nx^{n-1}$, we get $$\frac{dy}{dx} = 5 \times 5x^{5-1} = 25x^4.$$ 1b. Given $y = 24x^{3.5}$, find $\frac{dy}{dx}$. Using the power rule, $$\frac{dy}{dx} = 24 \times 3.5 x^{3.5-1} = 84x^{2.5}.$$ 1c. Given $y = \frac{1}{x} = x^{-1}$, find $\frac{dy}{dx}$. Using the power rule, $$\frac{dy}{dx} = -1 \times x^{-2} = -x^{-2} = -\frac{1}{x^2}.$$ 2a. Given $y = -\frac{4}{x^2} = -4x^{-2}$, find $\frac{dy}{dx}$. Using the power rule, $$\frac{dy}{dx} = -4 \times (-2) x^{-3} = 8x^{-3} = \frac{8}{x^3}.$$ 2b. Given $y = 2x$, find $\frac{dy}{dx}$. Derivative of $x$ is 1, so $$\frac{dy}{dx} = 2.$$ 3a. Given $y = 2\sqrt{x} = 2x^{1/2}$, find $\frac{dy}{dx}$. Using the power rule, $$\frac{dy}{dx} = 2 \times \frac{1}{2} x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}}.$$ 3b. Given $y = 3 \times 8\sqrt{x^5} = 3 \times 8 x^{5/8} = 24 x^{5/8}$, find $\frac{dy}{dx}$. Using the power rule, $$\frac{dy}{dx} = 24 \times \frac{5}{8} x^{5/8 - 1} = 15 x^{-3/8} = \frac{15}{x^{3/8}}.$$ 4a. Given $y = \frac{e^2 - e^{-x}}{2}$, find $\frac{dy}{dx}$. Derivative of $e^2$ is 0 (constant), derivative of $e^{-x}$ is $-e^{-x}$, so $$\frac{dy}{dx} = \frac{0 - (-e^{-x})}{2} = \frac{e^{-x}}{2}.$$ 5. Given $y = 1 - \frac{\sqrt{x}}{x} = 1 - \frac{x^{1/2}}{x} = 1 - x^{-1/2}$, find $\frac{dy}{dx}$. Derivative of 1 is 0, derivative of $-x^{-1/2}$ is $$- \left(-\frac{1}{2} x^{-3/2}\right) = \frac{1}{2} x^{-3/2} = \frac{1}{2x^{3/2}}.$$ 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. $\frac{15}{x^{3/8}}$ 4a. $\frac{e^{-x}}{2}$ 5. $\frac{1}{2x^{3/2}}$