1. **Problem:** Find the first and second derivatives of the function $f(x) = x^6$. 2. **Formula:** The derivative of $x^n$ with respect to $x$ is given by the power rule: $$f'(x) = nx^{n-1}$$ 3. **First derivative:** Applying the power rule to $f(x) = x^6$: $$f'(x) = 6x^{6-1} = 6x^5$$ 4. **Second derivative:** Differentiate $f'(x) = 6x^5$ again: $$f''(x) = 6 \times 5 x^{5-1} = 30x^4$$ 5. **Answer:** - $f'(x) = 6x^5$ - $f''(x) = 30x^4$