1. **Problem Statement:** Find the derivative $f'(x)$ for the function $f(x) = x + \sqrt{x}$. 2. **Recall the derivative rules:** The derivative of $x$ is 1. For $\sqrt{x}$, rewrite as $x^{1/2}$ and use the power rule: $\frac{d}{dx} x^n = n x^{n-1}$. 3. **Apply the derivative:** $$f'(x) = \frac{d}{dx} x + \frac{d}{dx} x^{1/2} = 1 + \frac{1}{2} x^{-1/2} = 1 + \frac{1}{2\sqrt{x}}.$$ 4. **Problem Statement:** Find $f'(x)$ for $f(x) = 2 + \frac{1}{x} + \frac{1}{x^2}$. 5. **Rewrite terms:** $\frac{1}{x} = x^{-1}$ and $\frac{1}{x^2} = x^{-2}$. 6. **Apply power rule:** $$f'(x) = 0 - x^{-2} - 2x^{-3} = -\frac{1}{x^2} - \frac{2}{x^3}.$$ 7. **Problem Statement:** Find $f'(x)$ for $f(x) = \sqrt{x} \cos(x)$. 8. **Use product rule:** $\frac{d}{dx} [u v] = u' v + u v'$, where $u = \sqrt{x} = x^{1/2}$ and $v = \cos(x)$. 9. **Derivatives:** $u' = \frac{1}{2} x^{-1/2}$, $v' = -\sin(x)$. 10. **Apply product rule:** $$f'(x) = \frac{1}{2} x^{-1/2} \cos(x) - x^{1/2} \sin(x).$$ 11. **Problem Statement:** Find $f'(x)$ for $f(x) = x^3 e^x$. 12. **Use product rule:** $u = x^3$, $v = e^x$, $u' = 3x^2$, $v' = e^x$. 13. **Apply product rule:** $$f'(x) = 3x^2 e^x + x^3 e^x = e^x (3x^2 + x^3).$$ 14. **Problem Statement:** Find $f'(x)$ for $f(x) = \frac{x^3}{e^x}$. 15. **Rewrite:** $f(x) = x^3 e^{-x}$. 16. **Use product rule:** $u = x^3$, $v = e^{-x}$, $u' = 3x^2$, $v' = -e^{-x}$. 17. **Apply product rule:** $$f'(x) = 3x^2 e^{-x} - x^3 e^{-x} = e^{-x} (3x^2 - x^3).$$ 18. **Problem Statement:** Find $f'(x)$ for $f(x) = (3x^2 + 2x + 1)^5$. 19. **Apply chain rule:** Let $u = 3x^2 + 2x + 1$, then $f(x) = u^5$. 20. **Derivatives:** $g(u) = u^5$, so $g'(u) = 5u^4$. $u'(x) = 6x + 2$. 21. **Apply chain rule:** $$f'(x) = g'(u) \cdot u'(x) = 5(3x^2 + 2x + 1)^4 (6x + 2).$$ **Final answers:** 1) $f'(x) = 1 + \frac{1}{2\sqrt{x}}$ 2) $f'(x) = -\frac{1}{x^2} - \frac{2}{x^3}$ 3) $f'(x) = \frac{1}{2} x^{-1/2} \cos(x) - x^{1/2} \sin(x)$ 4) $f'(x) = e^x (3x^2 + x^3)$ 5) $f'(x) = e^{-x} (3x^2 - x^3)$ 6) $f'(x) = 5(3x^2 + 2x + 1)^4 (6x + 2)$