1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $y = 8x + \frac{1}{x}$.\n\n2. **Recall the derivative rules:**\n- The derivative of $x^n$ is $nx^{n-1}$.\n- The derivative of a constant times a function is the constant times the derivative of the function.\n- The derivative of $\frac{1}{x}$ can be rewritten as $x^{-1}$, and then differentiated using the power rule.\n\n3. **Rewrite the function:**\n$$y = 8x + x^{-1}$$\n\n4. **Differentiate term-by-term:**\n- Derivative of $8x$ is $8$.\n- Derivative of $x^{-1}$ is $-1 \cdot x^{-2} = -\frac{1}{x^2}$.\n\n5. **Combine the results:**\n$$\frac{dy}{dx} = 8 - \frac{1}{x^2}$$\n\n6. **Final answer:**\nThe derivative of $y = 8x + \frac{1}{x}$ is $$\boxed{\frac{dy}{dx} = 8 - \frac{1}{x^2}}.$$