1. **Problem:** Find the derivative $f'(x)$ of the function $f(x) = x^6$. 2. **Formula:** Use the power rule for derivatives: $$\frac{d}{dx} x^n = n x^{n-1}$$ where $n$ is any real number. 3. **Apply the rule:** Here, $n=6$, so $$f'(x) = 6 x^{6-1} = 6 x^5$$ 4. **Explanation:** The power rule states that to differentiate $x^n$, multiply by the exponent and then reduce the exponent by one. 5. **Final answer:** $$\boxed{f'(x) = 6 x^5}$$