Here are the solutions for the differentiation problems: a) Given the function: $$y = \frac{3x+1}{x}$$ Step 1: Rewrite the function by splitting the fraction. $$y = \frac{3x}{x} + \frac{1}{x}$$ $$y = 3 + x^{-1}$$ Step 2: Differentiate each term with respect to $x$. Recall that $\frac{d}{dx}(c) = 0$ for a constant $c$, and $\frac{d}{dx}(x^n) = nx^{n-1}$. $$\frac{dy}{dx} = \frac{d}{dx}(3) + \frac{d}{dx}(x^{-1})$$ $$\frac{dy}{dx} = 0 + (-1)x^{-1-1}$$ $$\frac{dy}{dx} = -x^{-2}$$ Step 3: Express the answer with a positive exponent. $$\frac{dy}{dx} = -\frac{1}{x^2}$$ The derivative is $\boxed{-\frac{1}{x^2}}$. b) Given the function: $$y = 5x^2 - \frac{1}{x}$$ Step 1: Rewrite the second term using a negative exponent. $$y = 5x^2 - x^{-1}$$ Step 2: Differentiate each term with respect to $x$. $$\frac{dy}{dx} = \frac{d}{dx}(5x^2) - \frac{d}{dx}(x^{-1})$$ $$\frac{dy}{dx} = (2)(5)x^{2-1} - (-1)x^{-1-1}$$ $$\frac{dy}{dx} = 10x - (-1)x^{-2}$$ $$\frac{dy}{dx} = 10x + x^{-2}$$ Step 3: Express the answer with a positive exponent. $$\frac{dy}{dx} = 10x + \frac{1}{x^2}$$ The derivative is $\boxed{10x + \frac{1}{x^2}}$.