1a. Differentiate $y=5x^5$ using the power rule $\frac{d}{dx}x^n = nx^{n-1}$. $$\frac{dy}{dx} = 5 \times 5x^{5-1} = 25x^4$$ 1b. Differentiate $y=24x^{3.5}$ similarly. $$\frac{dy}{dx} = 24 \times 3.5 x^{3.5-1} = 84x^{2.5}$$ 1c. Rewrite $y=\frac{1}{x}$ as $y=x^{-1}$ and differentiate. $$\frac{dy}{dx} = -1 \times x^{-2} = -x^{-2} = -\frac{1}{x^2}$$ 2a. Rewrite $y=-\frac{4}{x^2}$ as $y=-4x^{-2}$ and differentiate. $$\frac{dy}{dx} = -4 \times (-2) x^{-3} = 8x^{-3} = \frac{8}{x^3}$$ 2b. Differentiate $y=2x$. $$\frac{dy}{dx} = 2$$ 3a. Rewrite $y=2\sqrt{x}$ as $y=2x^{1/2}$ and differentiate. $$\frac{dy}{dx} = 2 \times \frac{1}{2} x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}}$$ 3b. Rewrite $y=3^{8}\sqrt{x^5}$ as $y=3^{8} x^{5/2}$ and differentiate. $$\frac{dy}{dx} = 3^{8} \times \frac{5}{2} x^{\frac{5}{2}-1} = 3^{8} \times \frac{5}{2} x^{3/2}$$ 4a. Differentiate $y=\frac{e^{2} - e^{-x}}{2}$. Since $e^{2}$ is constant, derivative is zero. Derivative of $-e^{-x}$ is $-(-1)e^{-x} = e^{-x}$. So, $$\frac{dy}{dx} = \frac{0 + e^{-x}}{2} = \frac{e^{-x}}{2}$$ 4b. Rewrite $y=1 - \frac{\sqrt{x}}{x}$ as $y=1 - \frac{x^{1/2}}{x} = 1 - x^{1/2 - 1} = 1 - x^{-1/2}$. Differentiate: $$\frac{dy}{dx} = 0 - (-\frac{1}{2}) x^{-3/2} = \frac{1}{2} x^{-3/2} = \frac{1}{2x^{3/2}}$$