1. **State the problem:** We need to find the derivative $f'(x)$ of the function $$f(x) = \frac{1 + x^8}{5x}$$. 2. **Rewrite the function:** To differentiate easily, rewrite the function as a product with $x$ in the denominator expressed as $x^{-1}$: $$f(x) = \frac{1 + x^8}{5x} = \frac{1 + x^8}{5} \cdot x^{-1} = \frac{1}{5} (1 + x^8)x^{-1}$$. 3. **Apply the product:** Expand $f(x)$: $$f(x) = \frac{1}{5} (x^{-1} + x^8 \cdot x^{-1}) = \frac{1}{5} (x^{-1} + x^{7})$$. 4. **Differentiate using sum and power rules:** $$f'(x) = \frac{1}{5} \left( \frac{d}{dx} x^{-1} + \frac{d}{dx} x^{7} \right)$$ 5. **Calculate each derivative:** - $$\frac{d}{dx} x^{-1} = -1 \cdot x^{-2} = -x^{-2}$$ - $$\frac{d}{dx} x^{7} = 7x^{6}$$ 6. **Combine results:** $$f'(x) = \frac{1}{5} (-x^{-2} + 7x^{6}) = \frac{1}{5} \left( -\frac{1}{x^{2}} + 7x^{6} \right)$$. 7. **Final answer:** $$\boxed{f'(x) = \frac{7x^{6}}{5} - \frac{1}{5x^{2}}}$$