1. **Problem:** Find the derivative of the function $f(x) = \frac{8}{x} - x$. 2. **Formula:** The derivative of a function $f(x)$ is given by $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ (first principle of derivatives). 3. **Step 1:** Rewrite $f(x)$ as $f(x) = 8x^{-1} - x$ to apply power rule easily. 4. **Step 2:** Differentiate each term separately using the power rule $\frac{d}{dx} x^n = n x^{n-1}$. $$\frac{d}{dx} 8x^{-1} = 8 \times (-1) x^{-2} = -8x^{-2}$$ $$\frac{d}{dx} (-x) = -1$$ 5. **Step 3:** Combine the derivatives: $$f'(x) = -8x^{-2} - 1 = -\frac{8}{x^2} - 1$$ 6. **Final answer:** $$\boxed{f'(x) = -\frac{8}{x^2} - 1}$$ This derivative tells us the rate of change of the function $f(x)$ at any point $x$ except where $x=0$ (since division by zero is undefined).