3.3: Differentiation Rules

1. Given $f(x) = 14$, find $f'(x)$.
    
2. Given $g(x) = x^5$, find $g'(x)$.
    
3. Given $h(x) = \pi^3$, find $h'(x)$.
    
4. Given $y = 2x^3 + 7$, find $y'$.
    
5. Given $f(x) = 3x^2 - x + 4$, find $\dfrac{df}{dx}$.
    
6. Given $g(x) = \sqrt{2}x^5 - x^{\pi}$, find $\dfrac{dg}{dx}$.
    
7. Given $h(x) = \dfrac{1}{2}(x^3 + x^2 + x + 1)$, find $\dfrac{dh}{dx}$.
    
8. Given $y = \dfrac{2x^3 - 4}{3}$, find $\dfrac{dy}{dx}$.
    
9. Given $f(x) = 2x^{-3} + 4$, find $f'(x)$.
    
10. Given $g(x) = \dfrac{2}{x} + \dfrac{x}{2}$, find $g'(x)$.
     
11. Given $h(x) = \sqrt{x} - \dfrac{1}{\sqrt{x}}$, find $h'(x)$.
     
12. Given $y = \dfrac{1}{2x^{3/2}} + 2\sqrt{x}$, find $y'(x)$.
     
13. Given $f(x) = \sqrt[\leftroot{5} \uproot{2} \scriptstyle 3]{\dfrac{8}{x^2}}$, find $\dfrac{df}{dx}$.
     
14. Given $g(t) = (t + 1)^2$, find $\dfrac{dg}{dt}$.
     
15. Given $h(t) = -5\left(t^3 - 3t^2 - \dfrac{1}{t}\right)$, find $\dfrac{dh}{dt}$.
     
16. Given $x = \dfrac{t^2 + 1}{t}$, find $\dfrac{dx}{dt}$.
     
17. Given $f(x) = (x^2 + 2x)(2x^3 + 3x^2 - 4)$, find $f'(x)$.
     
18. Given $g(x) = (1 + x + x^2)(x^{-1} + x^{-2})$, find $g'(x)$.
     
19. Given $h(x) = \left(\dfrac{2}{x^3} - \dfrac{1}{x^5}\right)(x^2 - 5x + 1)$, find $h'(x)$.
     
20. Given $y = \dfrac{1}{x^2 + 1}$, find $y'$.
     
21. Given $f(x) = \dfrac{x^2 + \sqrt{x}}{x^2 - 1}$, find $\dfrac{df}{dx}$.
     
22. Given $g(x) = 3 - \dfrac{2x^{-2} - 4}{x^4}$, find $\dfrac{dg}{dx}$.
     
23. Given $h(x) = \dfrac{(x^2 + 1)(\sqrt{x} - 2)}{2x + 3}$, find $\dfrac{dh}{dx}$.
     
24. Given $y = (1 - \sqrt{x})\left(\dfrac{x}{x + 1}\right)$, find $\dfrac{dy}{dx}$.
     
25. Given $f(x) = |x|$, find $f'(x)$. [Hint: Consider first the case where $x > 0$, then separately consider the case where $x < 0$. For each case, consider how $|x|$ might be simplified.]
     
26. Find the equation of the line tangent to the graph of $g(x) = \dfrac{1}{x}$ at $x = \dfrac{1}{2}$.
     
27. Find the equation of the line tangent to the graph of $h(x) = \dfrac{x}{1 + x}$ at $x = 1$.
     
28. Given $f(x) = \dfrac{1}{6}x^3 - \dfrac{1}{2}x^2 + x - 4$, find $f'(x)$ and $f''(x)$.
     
29. Given $y = \dfrac{x + 1}{x - 1}$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$.