Differentiate the following functions: y = x5, y = 2x3, and y = 3x2 + 2x + 5. Show your work using the power rule and other relevant differentiation rules.

Using the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$:

$$\frac{dy}{dx} = 5x^{5-1}$$ $$\frac{dy}{dx} = 5x^4$$ $$\boxed{\frac{dy}{dx} = 5x^4}$$

Using the constant multiple rule $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$ and the power rule:

$$\frac{dy}{dx} = 2 \cdot (3x^{3-1})$$ $$\frac{dy}{dx} = 2 \cdot 3x^2$$ $$\frac{dy}{dx} = 6x^2$$ $$\boxed{\frac{dy}{dx} = 6x^2}$$

Using the sum rule $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))$ and the power rule, and the rule for a constant $\frac{d}{dx}(c) = 0$:

$$\frac{dy}{dx} = \frac{d}{dx}(3x^2) + \frac{d}{dx}(2x) + \frac{d}{dx}(5)$$ $$\frac{dy}{dx} = 3 \cdot (2x^{2-1}) + 2 \cdot (1x^{1-1}) + 0$$ $$\frac{dy}{dx} = 6x^1 + 2x^0$$ $$\frac{dy}{dx} = 6x + 2 \cdot 1$$ $$\frac{dy}{dx} = 6x + 2$$ $$\boxed{\frac{dy}{dx} = 6x + 2}$$