1. **Differentiate** $y = x^2$. Using the power rule $\frac{d}{dx} (x^n) = nx^{n-1}$, we get $$y' = 2x^{2-1} = 2x.$$ \boxed{y' = 2x} 2. **Differentiate** $y = 3x^3 + 2x$. Apply the power rule to each term: $$y' = 3 \cdot 3x^{3-1} + 2 \cdot 1x^{1-1} = 9x^2 + 2.$$ \boxed{y' = 9x^2 + 2} 3. **Differentiate** $y = \frac{1}{x}$. Rewrite as $y = x^{-1}$. Using the power rule: $$y' = -1 \cdot x^{-1-1} = -x^{-2} = -\frac{1}{x^2}.$$ \boxed{y' = -\frac{1}{x^2}} 4. **Differentiate** $y = \sqrt{x}$. Rewrite as $y = x^{\frac{1}{2}}$. Using the power rule: $$y' = \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}.$$ \boxed{y' = \frac{1}{2\sqrt{x}}} 5. **Differentiate** $y = (x + 1)^2$. Use the chain rule: derivative of outer function times derivative of inner function $$y' = 2(x + 1)^{2-1} \cdot 1 = 2(x + 1).$$ \boxed{y' = 2(x + 1)} 6. **Find the integral** $\int x \, dx$. Use the power rule for integrals: $$\int x \, dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C.$$ \boxed{\frac{x^2}{2} + C} 7. **Find the integral** $\int 2x^2 \, dx$. Factor out constant and apply power rule: $$\int 2x^2 \, dx = 2 \int x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} + C = \frac{2}{3} x^3 + C.$$ \boxed{\frac{2}{3} x^3 + C}