Determine whether each of the following statements is true or false, and explain why.

1. The derivative of \(x^x\) is \(\pi x^{n-1}\).

2. The derivative of a sum is the sum of the derivatives.

3. The derivative of a product is the product of the derivatives.

4. The derivative of a quotient is the quotient of the derivatives.

5. The chain rule is used to take the derivative of a product of functions.

6. The only function that is its own derivative is \(e^x\).

7. The derivative of \(10^x\) is \(\times 10^{x-1}\).

8. The derivative of \(\ln |x|\) is the same as the derivative of \(\ln x\).

9. The derivative of \(\ln kx\) is the same as the derivative of \(\ln x\).

10. The derivative of \(\log x\) is the same as the derivative of \(\ln x\).

11. \(D, \tan(x^2) = \sec^2(x^2)\).

12. \(D, \tan^2 x = 2\tan x \sec^2 x\).

Use the rules for derivatives to find the derivative of each function defined as follows:

13. \(y = 5x^3 - 7x^2 - 9x + 5\).

14. \(y = x^2 + 13x^5 - 4x^4 + 9x^3 - 6x + 4\).

15. \(y = 9x^{mn}\).

16. \(y = -4x^{-1}\).

17. \(f(x) = 3x^{-1} + 6x\).

18. \(f(x) = 19x^{-1} - 8x\).

19. \(L(x) = \frac{4x+7}{3x}\).

20. \(y = 3x^2 + 2x + x - 1 + x - 2\).

21. \(y = \frac{x+2}{2x^3} - 5x^2\).

22. \(f(x) = (3x^2 - 2)^4\).

23. \(y = x + \frac{5}{x^2}\).

24. \(y = \frac{2f}{7} - 5\).

25. \(y = \frac{3}{x^2} + 4x^3\).

26. \(y = 3x(2x+1)^3\).

27. \(r(t) = \frac{(3t+1)^3}{5t^2 - 7t}\).

28. \(y = x^x + \frac{x^2}{x}\).

29. \(p(t) = t^2 (t^2 + 1)^{3/2}\).

30. \(y = x^{40} + 4\sec x - 3\log x\).

31. \(y = -6e^{2x}\).

32. \(y = \sin^3 x + \cos^3 x\).

33. \(y = e^{-2n}\).

34. \(y = 3\sin^2(\cos 4x)\).

35. \(y = 5xe^{2x}\).

36. \(y = \left(x + \frac{x^2}{3}\right)^7\).

37. \(y = \ln(2 + x^2)\).

38. \(y = \frac{9 - 4x}{x}\).

39. \(y = \frac{x - 3}{\ln|3x|}\).

40. \(y = \frac{x}{x \cdot x}\).

41. \(y = \ln(x^2 - 1) \cdot xe^2\).

42. \(y = 2\sin(\sin(\sin 3x))\).

43. \(s = (t^2 + e^2)^2\).

44. \(y = \cos x + 2\cos x\).

45. \(y = 3 \cdot 10^{-x^2}\).

46. \(q = (e^{2p+1} - 2)^4\).

47. \(g(z) = \log_2(z^3 + z + 1)\).

48. \(h(z) = \log(1 + e)\).

49. \(f(x) = e^{2x} \ln(xe^x + 1)\).

50. \(f(x) = \ln(x + 1) e^x\).

51. \(y = 2\tan 5x\).

52. \(y = -4\sin 7x\).

53. \(y = \cot(6 - 3x^2)\).

54. \(y = \sin x\).

55. \(f(x) = 3x^{-1} + 6x\).

56. \(y = \frac{\sin x}{x}\).

57. \(y = 2\sin^4(4x^2)\).

58. \(y = x \sec x\).

59. \(y = \cos(1 + x^2)\).

60. \(y = \cot\left(\frac{21}{x^4}\right)\).

61. \(y = e^{-2x} \sin x\).

62. \(y = \tan^2(x^3)\).

63. \(y = \frac{1 - \cos x}{\cos^2 x}\).

64. \(y = \frac{1 - \sin 2x}{1 + \sin 2x}\).

65. \(y = \frac{1 + x}{\tan x}\).

66. \(y = \frac{\sec x}{6 - x}\).

67. \(y = \ln|5\sin x|\).

68. \(y = \ln|\cos x|\).